View source for Ideal

A special type of subobject of an algebraic structure. The concept of an ideal first arose in the theory of rings. The name ideal derives from the concept of an [[Ideal number|ideal number]].

For an algebra, a ring or a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500301.png" />, an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500302.png" /> is a subalgebra, subring or sub-semi-group closed under multiplication by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500303.png" />. Here an ideal is said to be a left (or right) ideal if it is closed under multiplication on the left (or right) by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500304.png" />, that is, if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500305.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500306.png" /></td> </tr></table>

An ideal that is simultaneously a left ideal and a right ideal (that is, one that is preserved under multiplication by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500307.png" />) is said to be two-sided. These three concepts coincide in the commutative case. Every assertion about left ideals has a corresponding dual assertion about right ideals (subsequent statements will refer only to the  "left case" ).

Two-sided ideals in rings and algebras play exactly the same role as do normal subgroups (cf. [[Normal subgroup|Normal subgroup]]) in groups. For every homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500308.png" />, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i0500309.png" /> (that is, the set of elements mapped to 0 by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003010.png" />) is an ideal, and conversely every ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003011.png" /> is the kernel of some homomorphism. Moreover, an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003012.png" /> determines a unique [[Congruence (in algebra)|congruence (in algebra)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003014.png" /> of which it is the zero class, and thus determines the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003015.png" /> of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003016.png" /> of which it is the kernel uniquely (up to an isomorphism): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003017.png" /> is isomorphic to the quotient ring or quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003018.png" />, denoted also by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003019.png" />. Ideals of multi-operator groups have similar properties in relation to homomorphisms. In a multi-operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003020.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003021.png" /> an ideal is defined to be a normal subgroup of its additive group satisfying the following property: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003022.png" />-ary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003023.png" />, arbitrary elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003025.png" />, the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003026.png" /></td> </tr></table>

holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003027.png" />. (This concept reduces to that of a two-sided ideal for rings and algebras.)

On the other hand, the two-sided ideals of a semi-group do not give a description of all homomorphic images of the semi-group. If a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003028.png" /> of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003029.png" /> onto a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003030.png" /> is given, then only in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003031.png" /> is a semi-group with zero it is possible to associate with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003032.png" /> a two-sided ideal in a natural way, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003033.png" />; however, this association need not determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003034.png" /> uniquely. Nevertheless, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003035.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003036.png" />, then among the quotient semi-groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003037.png" /> having the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003038.png" /> as an element there exists a maximal one, written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003039.png" /> (and called the ideal quotient). The elements of this semi-group are the elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003040.png" /> and the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003041.png" /> itself, which is the zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003042.png" />.

For an arbitrary subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003043.png" /> one can define the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003044.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003045.png" /> as the intersection of all ideals that contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003046.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003047.png" /> is said to be a basis of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003048.png" />. Different bases can generate one and the same ideal. An ideal generated by a single element is said to be a [[Principal ideal|principal ideal]].

The intersection, and for semi-groups also the union, of left (two-sided) ideals is again a left (two-sided) ideal. For rings and algebras, the set-theoretical union of ideals need not be an ideal. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003050.png" /> be left or two-sided ideals in a ring (or algebra) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003051.png" />. The sum of the ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003053.png" /> is the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003054.png" />; it is the smallest ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003055.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003057.png" />. The set of all (left or two-sided) ideals of a ring (or algebra) forms a lattice under the operations of intersection and taking sums. Many classes of rings and algebras are defined by conditions on their ideals or on the lattice of ideals (see [[Principal ideal ring|Principal ideal ring]]; [[Artinian ring|Artinian ring]]; [[Noetherian ring|Noetherian ring]]).

An ideal of the multiplicative semi-group of a ring may or may not be an ideal of the ring. A semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003058.png" /> is a group if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003059.png" /> has no (left or two-sided) ideal other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003060.png" />. Thus, the abundance of ideals in a semi-group characterizes the degree to which the semi-group differs from a group.

For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003061.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003062.png" /> (an algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003063.png" />), an ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003064.png" /> need not be an ideal of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003065.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003066.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003067.png" />-algebra with zero multiplication, the set of ideals of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003068.png" /> is the set of subgroups of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003069.png" />, while the set of ideals of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003070.png" /> is the set of all subspaces of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003071.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003072.png" />. However, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003073.png" /> is an algebra with identity, these concepts of an ideal coincide. Therefore many results have identical statements for rings and algebras.

A ring not having any two-sided ideal is said to be a [[Simple ring|simple ring]]. A ring without proper one-sided ideals is a [[Skew-field|skew-field]]. Left ideals of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003074.png" /> may also be defined as submodules of the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003075.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003076.png" />. Some properties of rings remain unchanged when right ideals are substituted for left ideals. For example, the [[Jacobson radical|Jacobson radical]] defined in terms of left ideals is the same as the Jacobson radical defined in terms of right ideals. On the other hand, a left Noetherian ring can fail to be right Noetherian.

The study of ideals in commutative rings is an important part of commutative algebra. With every commutative ring with identity one can associate the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003077.png" /> whose elements are the proper prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003078.png" />. There is a one-to-one correspondence between the set of all radicals of ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003079.png" /> and the set of closed subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003080.png" />.

The concept of an ideal of a field occurs in commutative algebra, more precisely, that of an ideal of a field relative to a ring. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003081.png" /> is a commutative ring with identity and without zero divisors, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003082.png" /> is the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003083.png" />. An ideal of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003084.png" /> is a non-zero subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003085.png" /> that is a subgroup of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003086.png" /> closed under multiplication by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003087.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003088.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003090.png" />) and such that there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003091.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003092.png" />. An ideal is said to be an integral ideal if it is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003093.png" /> (and then it is an ordinary ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003094.png" />); otherwise it is a [[Fractional ideal|fractional ideal]].

An ideal of a lattice is a non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003095.png" /> of a lattice such that: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003096.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003097.png" />; and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003098.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i05003099.png" />. A dual ideal (or a filter) of a lattice is defined in the dual manner (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030100.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030101.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030102.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030103.png" />). The ideals of a lattice also form a lattice under inclusion. A maximal element of the set of all proper ideals of a lattice is called a maximal ideal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030104.png" /> is a homomorphism of a lattice onto a partially ordered set with a zero, then the complete inverse image of the zero is an ideal. It is called the kernel ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030105.png" />. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030106.png" /> of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030107.png" /> is said to be a standard ideal if for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030109.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030110.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030111.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030113.png" />. Every standard ideal is a kernel ideal. A kernel ideal of a relatively complemented lattice (see [[Lattice with complements|Lattice with complements]]) is standard. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030114.png" /> is called a prime ideal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030115.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030116.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030117.png" />. Each of the following conditions is equivalent to primality for an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030118.png" /> of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030119.png" />: a) the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030120.png" /> is a filter; or b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030121.png" /> is the complete inverse image of zero under some homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030122.png" /> onto a two-element lattice. Every maximal ideal of a distributive lattice is prime.

The concept of an ideal in a partially ordered set is not in full agreement with the preceding definition. In fact, instead of 1), a stronger condition is required to hold: For every subset of the ideal, the supremum (join) of the set (if it exists) is also in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030123.png" />.

An ideal object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030124.png" /> of a category with null morphisms is a [[Subobject|subobject]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030125.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030126.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030127.png" /> for some morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030128.png" />. This ideal can be identified with the set of all monomorphisms that are kernels of some morphism (see also [[Normal monomorphism|Normal monomorphism]]). The concept of a co-ideal object of a category is defined in the dual way. The concept of an ideal for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030129.png" />-groups is a special case of that of an ideal object in a category.

A left ideal of a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030130.png" /> is a class of morphisms containing, with every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030131.png" /> of it, all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030132.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030133.png" />, if these are defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030134.png" />. Right ideals of a category are defined in the dual way. A two-sided ideal is a class of morphisms that is both a left ideal and a right ideal.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.A. Skornyakov,   "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>



====Comments====
There is some disagreement about the correct definition of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030135.png" /> in a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030136.png" />. Instead of the definition given above, some authors would allow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030137.png" /> to be an arbitrary lower set (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030138.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030139.png" />); others require additionally that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030140.png" /> be directed (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030142.png" />, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030143.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030144.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030145.png" />). The latter definition has the advantage of agreeing with the usual one in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030146.png" /> is a lattice (or a join semi-lattice).

For a [[Boolean algebra|Boolean algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030147.png" />, a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030148.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030149.png" /> is an ideal in the lattice-theoretic sense if and only if it is an ideal of the [[Boolean ring|Boolean ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050030/i050030150.png" />. It was this equivalence which led M.H. Stone [[#References|[a1]]] to extend the use of the term  "ideal"  from rings to lattices. Since then, the study of ideals has played an important role in lattice theory, and particularly in the theory of distributive lattices.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.H. Stone,   "Topological representation of distributive lattices and Brouwerian logics"  ''Časopis Pešt. Mat. Fys.'' , '''67'''  (1937)  pp. 1–25</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.T. Johnstone,   "Stone spaces" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Jacobson,   "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR></table>