# Definition(s) > [!NOTE] Definition (Ring with Unity Axioms) >A *ring* is an [[Algebraic Structure|algebraic structure]] $(R,+,\times)$ where $R$ is a set and $+,\times$ are [[Binary Operation|binary operations]] on $R,$ which satisfy the following conditions: > >(A0) Closure under addition: $\forall a,b\in R:a+b\in R.$ > >(A1) [[Associativity|Associativity]] of [[Ring Addition|addition]]: $\forall a,b,c\in R:(a+b)+c=a+(b+c).$ > >(A2) [[Commutativity|Commutativity]] of addition: $\forall a,b \in R: a+b=b+a.$ > >(A3) Existence of Additive [[Identity element of a binary operation|Identity]]/ [[Ring Zero Element|Zero]] Element: $\exists 0_{R}\in R: \forall a\in R:a+0_{R}=a=0_{R}+a.$ > >(A4) Existence of additive [[Inverse under a binary operation|inverses]]/ Negative Elements: $\forall a\in R: \exists -a \in R:a+(-a)=0_{R}=-a+a$ > >(M0) Closure of [[Ring Product|multiplication]]: $\forall a,b\in R: a\times b \in R.$ > >(M1) Associativity of multiplication: $\forall a,b,c\in R:(a\times b)\times c=a\times(b\times c)$ > >(M2) Existence of a multiplicative identity/ [[Ring Unity|Unity]]: $\exists 1 \in R: \forall a:a\times 1=a = 1\times a$ > >(B) Multiplication is [[Distributivity|distributive]] over addition: $\forall a ,b,c \in R: (a\times(b+c)=a\times b +a\times c) \land((b+c)\times a=b\times a+c\times a)$ **Variations in definitions**: We can summarise (A0)-(A4) by asserting that $(R,+)$ is an [[Abelian Group|abelian group]] and (M0)-(M2) by asserting that $(R,\times)$ is a [[Monoid]]. Some sources define a ring (or *rng*, pronounced rung) as the algebraic structure satisfying the above axioms except (M2). > [!Example] Examples > The set of [[Integers|integers]] $\mathbb{Z}$ is a ring under integer addition and multiplication. The [[Unit in a Ring|units]] of $\mathbb{Z}$ are $1$ and $-1.$ > >The [[Zero Ring|zero ring]] is the ring with just one element. [[Zero Ring iff Unity Equals Zero|In this case and this case only, the unity and zero are the same]]. > >[[Ring of Polynomial Forms]] # Properties **Consequences of ring axioms:** Note that [[Ring Product With Zero is Zero|any ring product with zero is zero]]. By [[Product With Ring Negative|product with negative]], for all $a,b\in R,$ $-(ab)=a(-b)=(-a)b.$ A corollary is [[Product With Ring Negatives|that the product of ring negatives is 'positive']]. Therefore, if $R$ has a unity element $1$ then $(-1)a=-a$ and $(-1)(-1)=1.$ **Subrings & quotients:** A subset $S$ of a ring $R$ is [[Subring|subring]] of $R$ if $S$ is itself a ring under the operations of $R.$ An [[Ideal of Ring|ideal]] is a special type of subring: not only is the product of any two members of the ideal in the ideal, but every product involving at least one member of the ideal is in the ideal. For example $n\mathbb{Z}$ is an ideal of $\mathbb{Z}$ for any $n\in \mathbb{Z}.$ A subring is not necessarily ideal: $\mathbb{Z}$ is a non-ideal subring of $\mathbb{R}.$ Note that $\mathbb{Z}$ is still a normal subgroup of $(\mathbb{R},+)$ addition and so $\mathbb{R}/\mathbb{Z}$ is a [[Quotient Group|quotient group]]. The [[Quotient Ring|quotient ring]] of a ring by an ideal is defined as the coset space of the ideal under ... In this case, [[Quotient Ring Operations are Well-Defined|quotient ring operations are well-defined]] thus [[Quotient Ring is Ring|quotient ring is indeed a ring]]. **Comparing Rings**: [[Homomorphism of Rings]] **Types**: A [[Commutative Ring|commutative ring]] is a ring whose ring product is commutative. A [[Division Ring|division ring]] is a ring with unity whose non-zero elements each have a multiplicative inverse in the ring (every non-zero element is a [[Unit in a Ring|unit]]). A [[Field (Algebra)|field]] is a non-zero division ring that is commutative. A [[Zero Divisor|zero divisor]] is an element that can be pre or post multiplied by a non-zero element to give zero. An [[Integral Domain|integral domain]] is a non-zero commutative ring in which there are no non-zero proper divisors. # References 1. MA268