Difference between revisions of "Associative rings and algebras"
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− | + | {{MSC|16}} | |
+ | {{TEX|done}} | ||
− | + | ''Associative rings and algebras'' are | |
+ | rings and algebras with an associative multiplication, i.e., sets with | ||
+ | two binary operations, addition $+$ and multiplication $\cdot$, that are | ||
+ | Abelian groups with respect to addition and semi-groups with respect | ||
+ | to multiplication, and in which the multiplication is distributive | ||
+ | (from the left and from the right) with respect to the | ||
+ | addition. Moreover, an associative algebra should be a vector space | ||
+ | over a fixed field $F$, and the multiplication in it is compatible | ||
+ | with multiplication by field elements in the sense of the condition | ||
+ | $\alpha(ab) = (\alpha a)b = a(\alpha b)$ for all $\alpha\in F$ and all $a,b$ from the algebra. | ||
− | The | + | The first examples of associative rings and associative algebras were |
+ | number rings and fields (the field of complex numbers and its | ||
+ | subrings), polynomial algebras, matrix algebras over fields, and | ||
+ | function fields. The theory of associative rings and algebras became | ||
+ | an independent part of algebra at the beginning of the 20th | ||
+ | century. This theory has many contact points with numerous fields of | ||
+ | mathematics, especially so with algebraic geometry and algebraic | ||
+ | number theory (commutative rings), functional analysis (commutative | ||
+ | normed rings, operator rings and function rings), and topology (rings | ||
+ | of continuous functions on topological spaces). The theory of fields, | ||
+ | the theory of commutative rings (cf. | ||
+ | [[Field|Field]]; | ||
+ | [[Commutative ring|Commutative ring]], see also | ||
+ | [[Commutative algebra|Commutative algebra]]) and the theory of | ||
+ | representations of associative algebras have become independent | ||
+ | branches of the theory of associative rings and algebras. The theory | ||
+ | of topological rings and skew-fields forms part of | ||
+ | [[Topological algebra|topological algebra]]. | ||
− | + | The classical part of the theory of associative rings and algebras is | |
+ | formed by the theory of finite-dimensional associative algebras | ||
+ | {{Cite|Al}}. The principal results of this theory are: A | ||
+ | finite-dimensional simple associative algebra (i.e. one without proper | ||
+ | ideals) over a field $F$ is a complete matrix algebra over some | ||
+ | skew-field that is finite-dimensional over $F$ (Wedderburn's theorem); | ||
+ | a finite-dimensional associative algebra over a field of | ||
+ | characteristic zero (and, even more generally — a separable | ||
+ | finite-dimensional associative algebra) is the direct sum (as linear | ||
+ | spaces) of its radical I (i.e. its maximal nilpotent ideal) and some | ||
+ | semi-simple (i.e. with a radical zero) subalgebra $S$, any two | ||
+ | complementary semi-simple subalgebras $S$ and $S_1$ being conjugate (cf. | ||
+ | [[Wedderburn–Mal'tsev theorem|Wedderburn–Mal'tsev theorem]]). | ||
− | + | One of the most important classes of associative algebras are | |
+ | skew-fields (cf. | ||
+ | [[Skew-field|Skew-field]], i.e. associative rings in which the | ||
+ | equations $ax=b$ and $ya=b$ are solvable for all $a$, $b$ in the ring, | ||
+ | $a\ne 0$). Skew-fields that are algebras over a certain field are called | ||
+ | division algebras (cf. | ||
+ | [[Division algebra|Division algebra]]). The theory of | ||
+ | finite-dimensional division algebras forms a classical part of the | ||
+ | theory of fields. All finite-dimensional associative division algebras | ||
+ | over the field of real numbers have been described: these are the | ||
+ | field of real numbers, the field of complex numbers and the skew-field | ||
+ | of quaternions (cf. | ||
+ | [[Frobenius theorem|Frobenius theorem]]). All finite skew-fields are | ||
+ | commutative (Wedderburn's theorem on skew-fields). A Galois theory of | ||
+ | skew-fields has been constructed | ||
+ | {{Cite|Ja2}}. | ||
− | The theory of | + | The key concepts in the structural theory of associative rings are the |
+ | concepts of the | ||
+ | [[Jacobson radical|Jacobson radical]], semi-simplicity and | ||
+ | primitivity. An associative ring is called semi-simple (in the sense | ||
+ | of Jacobson) if its Jacobson radical is zero. A ring is called | ||
+ | primitive (from the right) if it has an irreducible exact right | ||
+ | module. All semi-simple associative rings are a subdirect sum of | ||
+ | primitive rings. Any primitive associative ring $R$ is a dense ring of | ||
+ | linear transformations of some vector space $V$ over a skew-field | ||
+ | (Jacobson's density theorem); here the meaning of density is that for | ||
+ | all linearly independent elements $v_1\dots,v_k$ from $V$ and all elements $u_1,\dots,u_k$ | ||
+ | from $V$ there exists a transformation $r\in R$ such that $v_ir = u_i$ for $1\le i\le k$. The | ||
+ | general theory of radicals occupies an important place in the | ||
+ | structural theory of rings (cf. | ||
+ | [[Radical of rings and algebras|Radical of rings and algebras]]). | ||
− | The | + | The theory of Artinian rings (from the right, cf. |
+ | [[Artinian ring|Artinian ring]]), i.e. rings with the descending chain | ||
+ | condition (minimum condition) for right ideals, forms a classical part | ||
+ | of the theory of associative rings. The main result of this theory is | ||
+ | that an associative ring is a semi-simple Artinian ring if and only if | ||
+ | it is a direct sum of a finite number of complete matrix rings over | ||
+ | skew-fields (the | ||
+ | [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]). | ||
− | + | The concept of the (classical) ring of fractions is important in the | |
+ | structural theory of associative rings. A ring $Q(R)$ is called the | ||
+ | (right) ring of fractions of a subring $R$ of it if in $Q(R)$ all the | ||
+ | regular elements (i.e. not divisors of zero) of $R$ are invertible and | ||
+ | if any element of $Q(R)$ has the form $ab^{-1}$, where $a,b\in R$. An associative ring | ||
+ | has a ring of fractions if and only if for all elements $a,b\in R$, where $B$ | ||
+ | is regular, there exist elements $a_1,b_1\in R$ such that $ab_1 = ba_1$ and $b_1$ is regular | ||
+ | (Ore's theorem). A ring $R$ has a semi-simple Artinian ring of | ||
+ | fractions if and only if it is semi-primary (i.e. $I^2 \ne 0$ for all non-zero | ||
+ | ideals $I\subset R$), satisfies the minimum condition for right annihilating | ||
+ | ideals of the form | ||
+ | $$r(S) = \{x\in R: Sx=0\},$$ | ||
+ | where $S$ is a subset of $R$, and contains no | ||
+ | infinite direct sums of right ideals (Goldie's theorem). In addition | ||
+ | to the classical rings of fractions, rings of fractions having other | ||
+ | meanings are also studied — mainly maximal or complete rings of | ||
+ | fractions | ||
+ | {{Cite|La}}. | ||
− | + | Special stress is laid on the study of free associative algebras (cf. | |
+ | [[Free associative algebra|Free associative algebra]]). Let $F$ be a | ||
+ | field and let $X$ be a set. The free associative algebra $\def\<{\langle}\def\>{\rangle}F\<X\>$ over $F$ | ||
+ | with a unit element and with base $X$ is the algebra of | ||
+ | non-commutative polynomials with free terms in the set of variables | ||
+ | $X$ and with coefficients from $F$. The algebra $F\<X\>$ is characterized | ||
+ | by the fact that it is generated by $X$ as an algebra with a unit | ||
+ | element and any mapping of $X$ into an associative algebra $R$ with a | ||
+ | unit element can be extended (even uniquely) to a homomorphism from | ||
+ | $F\<X\>$ into $R$. A free associative algebra is a ring with free ideals, | ||
+ | i.e. the right (left) ideals of the ring $F\<X\>$ are free right (left) | ||
+ | $F\<X\>$-modules, and all bases of the free finitely-generated $F\<X\>$-modules | ||
+ | contain the same number of elements (Cohn's theorem). Other examples | ||
+ | of rings with free ideals are group algebras of free groups, and free | ||
+ | products of associative division algebras. The free associative | ||
+ | algebra $F\<X\>$ is also a unique factorization domain: Any non-invertible | ||
+ | element $a\in F\<X\>$ has a representation $a=p_1\cdots p_n$, where $p_i$ are irreducible | ||
+ | elements, and this representation is unique except for the order of | ||
+ | the sequence and similarities (two elements $a$ and $b$ of a ring $R$ | ||
+ | are called similar if $R/aR$ and $R/bR$ are isomorphic as right | ||
+ | $R$-modules). The | ||
+ | [[Centralizer|centralizer]] of each non-scalar element of the algebra | ||
+ | $F\<X\>$ is isomorphic to the algebra of polynomials $F[t]$ in one variable | ||
+ | $t$ (Bergman's theorem). | ||
− | + | Group algebras and PI-algebras are important classes of associative | |
− | + | algebras (cf. | |
− | Group algebras and PI-algebras are important classes of associative algebras (cf. [[Group algebra|Group algebra]]; [[PI-algebra|PI-algebra]]). The theory of varieties of rings is in the process of being developed. | + | [[Group algebra|Group algebra]]; |
− | + | [[PI-algebra|PI-algebra]]). The theory of varieties of rings is in the | |
− | + | process of being developed. | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
+ | The importance of ring theory in mathematics has increased with the | ||
+ | development of | ||
+ | [[Homological algebra|homological algebra]]. Many known classes of | ||
+ | rings may be characterized in terms of properties of the category of | ||
+ | modules (cf. | ||
+ | [[Modules, category of|Modules, category of]]) over these rings. For | ||
+ | instance, a ring $R$ is a semi-simple Artinian ring if and only if all | ||
+ | right (left) modules over $R$ are projective (injective). A ring $R$ | ||
+ | is regular (in the sense of von Neumann) if and only if all right | ||
+ | (left) modules over $R$ are flat. See also | ||
+ | [[Regular ring (in the sense of von Neumann)|Regular ring (in the | ||
+ | sense of von Neumann)]]; | ||
+ | [[Homological classification of rings|Homological classification of | ||
+ | rings]]; | ||
+ | [[Quasi-Frobenius ring|Quasi-Frobenius ring]]. | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |- | ||
+ | |valign="top"|{{Ref|Al}}||valign="top"| A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) {{MR|0000595}} {{ZBL|0023.19901}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|ArNeTh}}||valign="top"| E. Artin, C.J. Nesbitt, R.M. Thrall, "Rings with minimum condition", Univ. Michigan Press, Ann Arbor (1946) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|CaEi}}||valign="top"| H. Cartan, S. Eilenberg, "Homological algebra", Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Co}}||valign="top"| P.M. Cohn, "Free rings and their relations", Acad. Press (1971) {{MR|0371938}} {{ZBL|0232.16003}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience (1962) {{MR|0144979}} {{ZBL|0131.25601}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Di}}||valign="top"| N.J. Divinsky, "Rings and radicals", Allen & Unwin (1965) {{MR|0197489}} {{MR|0184971}} {{MR|0175939}} {{ZBL|0142.27703}} {{ZBL|0138.26303}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Fa}}||valign="top"| C. Faith, "Algebra: rings, modules, and categories", '''1''', Springer (1973) {{MR|0366960}} {{ZBL|0266.16001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Fa2}}||valign="top"| C. Faith, "Algebra", '''II. Ring theory''', Springer (1976) {{MR|0427349}} {{ZBL|0335.16002}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Gi}}||valign="top"| R. Gilmore, "Multiplicative ideal theory", M. Dekker (1972) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|He}}||valign="top"| I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer. (1968) {{MR|0227205}} {{ZBL|0177.05801}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "The theory of rings", Amer. Math. Soc. (1943) {{MR|0008601}} {{ZBL|0060.07302}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ja2}}||valign="top"| N. Jacobson, "Structure of rings", Amer. Math. Soc. (1956) {{MR|0081264}} {{ZBL|0073.02002}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) {{MR|0158000}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|La}}||valign="top"| J. Lambek, "Lectures on rings and modules", Blaisdell (1966) {{MR|0206032}} {{ZBL|0143.26403}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Mc}}||valign="top"| B.R. McDonald, "Finite rings with identity", M. Dekker (1974) {{MR|0354768}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Pa}}||valign="top"| D.S. Passman, "Infinite group rings", New York (1971) {{MR|0314951}} {{ZBL|0247.16004}} {{ZBL|0221.20042}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Po}}||valign="top"| L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Pr}}||valign="top"| C. Procesi, "Rings with polynomial identities", M. Dekker (1973) {{MR|0366968}} {{ZBL|0262.16018}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', v. Nostrand (1960) {{MR|0120249}} {{ZBL|0121.27801}} | ||
+ | |- | ||
+ | |} |
Latest revision as of 20:09, 15 December 2020
2010 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]
Associative rings and algebras are rings and algebras with an associative multiplication, i.e., sets with two binary operations, addition $+$ and multiplication $\cdot$, that are Abelian groups with respect to addition and semi-groups with respect to multiplication, and in which the multiplication is distributive (from the left and from the right) with respect to the addition. Moreover, an associative algebra should be a vector space over a fixed field $F$, and the multiplication in it is compatible with multiplication by field elements in the sense of the condition $\alpha(ab) = (\alpha a)b = a(\alpha b)$ for all $\alpha\in F$ and all $a,b$ from the algebra.
The first examples of associative rings and associative algebras were number rings and fields (the field of complex numbers and its subrings), polynomial algebras, matrix algebras over fields, and function fields. The theory of associative rings and algebras became an independent part of algebra at the beginning of the 20th century. This theory has many contact points with numerous fields of mathematics, especially so with algebraic geometry and algebraic number theory (commutative rings), functional analysis (commutative normed rings, operator rings and function rings), and topology (rings of continuous functions on topological spaces). The theory of fields, the theory of commutative rings (cf. Field; Commutative ring, see also Commutative algebra) and the theory of representations of associative algebras have become independent branches of the theory of associative rings and algebras. The theory of topological rings and skew-fields forms part of topological algebra.
The classical part of the theory of associative rings and algebras is formed by the theory of finite-dimensional associative algebras [Al]. The principal results of this theory are: A finite-dimensional simple associative algebra (i.e. one without proper ideals) over a field $F$ is a complete matrix algebra over some skew-field that is finite-dimensional over $F$ (Wedderburn's theorem); a finite-dimensional associative algebra over a field of characteristic zero (and, even more generally — a separable finite-dimensional associative algebra) is the direct sum (as linear spaces) of its radical I (i.e. its maximal nilpotent ideal) and some semi-simple (i.e. with a radical zero) subalgebra $S$, any two complementary semi-simple subalgebras $S$ and $S_1$ being conjugate (cf. Wedderburn–Mal'tsev theorem).
One of the most important classes of associative algebras are skew-fields (cf. Skew-field, i.e. associative rings in which the equations $ax=b$ and $ya=b$ are solvable for all $a$, $b$ in the ring, $a\ne 0$). Skew-fields that are algebras over a certain field are called division algebras (cf. Division algebra). The theory of finite-dimensional division algebras forms a classical part of the theory of fields. All finite-dimensional associative division algebras over the field of real numbers have been described: these are the field of real numbers, the field of complex numbers and the skew-field of quaternions (cf. Frobenius theorem). All finite skew-fields are commutative (Wedderburn's theorem on skew-fields). A Galois theory of skew-fields has been constructed [Ja2].
The key concepts in the structural theory of associative rings are the concepts of the Jacobson radical, semi-simplicity and primitivity. An associative ring is called semi-simple (in the sense of Jacobson) if its Jacobson radical is zero. A ring is called primitive (from the right) if it has an irreducible exact right module. All semi-simple associative rings are a subdirect sum of primitive rings. Any primitive associative ring $R$ is a dense ring of linear transformations of some vector space $V$ over a skew-field (Jacobson's density theorem); here the meaning of density is that for all linearly independent elements $v_1\dots,v_k$ from $V$ and all elements $u_1,\dots,u_k$ from $V$ there exists a transformation $r\in R$ such that $v_ir = u_i$ for $1\le i\le k$. The general theory of radicals occupies an important place in the structural theory of rings (cf. Radical of rings and algebras).
The theory of Artinian rings (from the right, cf. Artinian ring), i.e. rings with the descending chain condition (minimum condition) for right ideals, forms a classical part of the theory of associative rings. The main result of this theory is that an associative ring is a semi-simple Artinian ring if and only if it is a direct sum of a finite number of complete matrix rings over skew-fields (the Wedderburn–Artin theorem).
The concept of the (classical) ring of fractions is important in the structural theory of associative rings. A ring $Q(R)$ is called the (right) ring of fractions of a subring $R$ of it if in $Q(R)$ all the regular elements (i.e. not divisors of zero) of $R$ are invertible and if any element of $Q(R)$ has the form $ab^{-1}$, where $a,b\in R$. An associative ring has a ring of fractions if and only if for all elements $a,b\in R$, where $B$ is regular, there exist elements $a_1,b_1\in R$ such that $ab_1 = ba_1$ and $b_1$ is regular (Ore's theorem). A ring $R$ has a semi-simple Artinian ring of fractions if and only if it is semi-primary (i.e. $I^2 \ne 0$ for all non-zero ideals $I\subset R$), satisfies the minimum condition for right annihilating ideals of the form $$r(S) = \{x\in R: Sx=0\},$$ where $S$ is a subset of $R$, and contains no infinite direct sums of right ideals (Goldie's theorem). In addition to the classical rings of fractions, rings of fractions having other meanings are also studied — mainly maximal or complete rings of fractions [La].
Special stress is laid on the study of free associative algebras (cf. Free associative algebra). Let $F$ be a field and let $X$ be a set. The free associative algebra $\def\<{\langle}\def\>{\rangle}F\<X\>$ over $F$ with a unit element and with base $X$ is the algebra of non-commutative polynomials with free terms in the set of variables $X$ and with coefficients from $F$. The algebra $F\<X\>$ is characterized by the fact that it is generated by $X$ as an algebra with a unit element and any mapping of $X$ into an associative algebra $R$ with a unit element can be extended (even uniquely) to a homomorphism from $F\<X\>$ into $R$. A free associative algebra is a ring with free ideals, i.e. the right (left) ideals of the ring $F\<X\>$ are free right (left) $F\<X\>$-modules, and all bases of the free finitely-generated $F\<X\>$-modules contain the same number of elements (Cohn's theorem). Other examples of rings with free ideals are group algebras of free groups, and free products of associative division algebras. The free associative algebra $F\<X\>$ is also a unique factorization domain: Any non-invertible element $a\in F\<X\>$ has a representation $a=p_1\cdots p_n$, where $p_i$ are irreducible elements, and this representation is unique except for the order of the sequence and similarities (two elements $a$ and $b$ of a ring $R$ are called similar if $R/aR$ and $R/bR$ are isomorphic as right $R$-modules). The centralizer of each non-scalar element of the algebra $F\<X\>$ is isomorphic to the algebra of polynomials $F[t]$ in one variable $t$ (Bergman's theorem).
Group algebras and PI-algebras are important classes of associative algebras (cf. Group algebra; PI-algebra). The theory of varieties of rings is in the process of being developed.
The importance of ring theory in mathematics has increased with the development of homological algebra. Many known classes of rings may be characterized in terms of properties of the category of modules (cf. Modules, category of) over these rings. For instance, a ring $R$ is a semi-simple Artinian ring if and only if all right (left) modules over $R$ are projective (injective). A ring $R$ is regular (in the sense of von Neumann) if and only if all right (left) modules over $R$ are flat. See also Regular ring (in the sense of von Neumann); Homological classification of rings; Quasi-Frobenius ring.
References
[Al] | A.A. Albert, "Structure of algebras", Amer. Math. Soc. (1939) MR0000595 Zbl 0023.19901 |
[ArNeTh] | E. Artin, C.J. Nesbitt, R.M. Thrall, "Rings with minimum condition", Univ. Michigan Press, Ann Arbor (1946) |
[CaEi] | H. Cartan, S. Eilenberg, "Homological algebra", Princeton Univ. Press (1956) MR0077480 Zbl 0075.24305 |
[Co] | P.M. Cohn, "Free rings and their relations", Acad. Press (1971) MR0371938 Zbl 0232.16003 |
[CuRe] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras", Interscience (1962) MR0144979 Zbl 0131.25601 |
[Di] | N.J. Divinsky, "Rings and radicals", Allen & Unwin (1965) MR0197489 MR0184971 MR0175939 Zbl 0142.27703 Zbl 0138.26303 |
[Fa] | C. Faith, "Algebra: rings, modules, and categories", 1, Springer (1973) MR0366960 Zbl 0266.16001 |
[Fa2] | C. Faith, "Algebra", II. Ring theory, Springer (1976) MR0427349 Zbl 0335.16002 |
[Gi] | R. Gilmore, "Multiplicative ideal theory", M. Dekker (1972) |
[He] | I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer. (1968) MR0227205 Zbl 0177.05801 |
[Ja] | N. Jacobson, "The theory of rings", Amer. Math. Soc. (1943) MR0008601 Zbl 0060.07302 |
[Ja2] | N. Jacobson, "Structure of rings", Amer. Math. Soc. (1956) MR0081264 Zbl 0073.02002 |
[Ku] | A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000 |
[La] | J. Lambek, "Lectures on rings and modules", Blaisdell (1966) MR0206032 Zbl 0143.26403 |
[Mc] | B.R. McDonald, "Finite rings with identity", M. Dekker (1974) MR0354768 |
[Pa] | D.S. Passman, "Infinite group rings", New York (1971) MR0314951 Zbl 0247.16004 Zbl 0221.20042 |
[Po] | L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian) |
[Pr] | C. Procesi, "Rings with polynomial identities", M. Dekker (1973) MR0366968 Zbl 0262.16018 |
[ZaSa] | O. Zariski, P. Samuel, "Commutative algebra", 1, v. Nostrand (1960) MR0120249 Zbl 0121.27801 |
Associative rings and algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Associative_rings_and_algebras&oldid=15123