Ring theory MOC

Rng

A rng rʊŋ is a generalized ring which may lack a multiplicative identity. That is, a rng consists of an Abelian group called addition and a Semigroup called multiplication, with the extra conditions ring

• left-distributivity
• right-distributivity

These are precisely the semigroup objects in Category of abelian groups.

Examples

An example of a rng that is not a ring is the even integers

with the ordinary operations of integer addition and multiplication.

Properties

Let and

4. and

Proof of 1–5

Clearly so , and likewise for , proving ^P1. Similarly and likewise for , proving ^P2. It follows that , proving ^P3. Note , and likewise for right-distributivity, proving ^P4.

Now

proving ^P5.

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