2.4. Rings

So far we have studied algebraic structures with only one operation. Now we study rings which are sets with two (compatible) binary operations. Unlike groups, these two operations are usually denoted by + and · . One can, of course, go for general notations for these operations. However, that generalization doesn’t seem to pay much, but complicates matters. We stick to the conventions.

2.4.1. Definition and Basic Properties

Definition 2.12.

A ring (R, +, ·) (or R in short) is a set R together with two binary operations + and · on R such that the following conditions are satisfied. As in the case of multiplicative groups we write ab for a · b.

  1. Additive group The set R is an Abelian group under +. The additive identity is denoted ...

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