View source for Ring

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A set $R$ on which two binary algebraic operations are defined: addition and multiplication, the set being an [[Abelian group|Abelian group]] (the additive group of the ring) with respect to addition, and the multiplication is related to the addition by the [[distributive law]]s:

$$a(b+c)=ab+ac,\quad(b+c)a=ba+ca,$$

where $a,b,c\in R$. In general no restriction is imposed on multiplication, that is, $R$ is a [[groupoid]] (called the multiplicative groupoid of the ring) with respect to multiplication.

A non-empty subset $A\subset R$ is called a subring of $R$ if $A$ itself is a ring with respect to the operations defined on $R$, that is, $A$ must be a subgroup of the additive group of $R$ and a subgroupoid of the multiplicative groupoid of this ring. Clearly, the ring itself and the zero subring consisting of just the zero element are subrings of a given ring. The (set-theoretic) intersection of subrings of a ring is a subring. The join of a family of subrings $A_\alpha$, $\alpha\in I$, of a ring $R$ is the intersection of all subrings that contain all $A_\alpha$. The set of all subrings of a given ring is a [[Lattice|lattice]], $S(R)$, with respect to the operations of intersection and join of subrings. The set of ideals (cf. [[Ideal]]) of this ring forms a sublattice of $S(R)$.

Concerning the various directions in the theory of rings, see [[Rings and algebras]]; [[Associative rings and algebras]]; [[Non-associative rings and algebras]].



====Comments====
In many contexts it is tacitly assumed that the ring contains a unit element, denoted by $1$, and subrings are taken to be subrings with the same unit. In this case the set of ideals is not a sublattice of $S(R)$.