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We are currently studying rings in my class and a requirement in the definition of a ring that my book offers maintains that under addition, R is an abelian group. What exactly does that mean? For some reason my professor started off with rings and not with groups so some of the definitions refer to group theory, assuming prior knowledge.

I guess Im also asking in general a more descriptive definition of a ring and its structures besides "a nonempty set with two operations, addition and multiplication"

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A lot of books/professors start with rings instead of groups because a lot of the proofs or properties of groups involve number theory, primes, irreducibles, euclidean algorithm, division algorithm, etc which follow from ring theory.

Specifically Dummit and Foote define a ring as follows:

A ring $R$ is a set together with two binary operations $+$ and $\times$ (called addition and multiplication) (which just means the operations are closed, so if $a, b \in R$, then $a + b \in R$ and $a \times b \in R$) satisfying the following axioms:

  1. $(R, +)$ is an abelian group.
  2. $\times$ is associative: $(a \times b) \times c = a \times(b \times c)$ for all $a, b, c \in R$.
  3. The distributive law holds in $R$: for all $a, b, c\in R$ $$(a + b) \times c = (a \times c) + (b \times c) \text{ and } a \times (b + c) = (a \times b) + (a \times c)$$

By saying $(R, +)$ is an abelian group, we mean there is an identity element $0$ such that $a + 0 = a$ for all $a \in R$, there is an additive inverse $-a$ such that $a + (-a) = 0$ for all $a \in R$ and addition is associative and commutative.

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  • $\begingroup$ Thanks a lot. That makes so much more sense now. $\endgroup$ – Gamecocks99 Feb 9 '13 at 8:55
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In abstract algebra, an abelian group $G$ , also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).

That is $a+b=b+a,\forall a,b\in G$

You'll find a lot of examples here

Ring definition and examples from wikipedia

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