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Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

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CHAPTER 9

Rings

1 Definition and examples

Definition. By a ring we mean a nonempty set R with two binary operations + and ·, called addition and multiplication (also called product), respectively, such that

(i) (R,+) is an additive abelian group.

(ii) (R, ·) is a multiplicative semigroup.

(iii) Multiplication is distributive (on both sides) over addition; that is, for all images,

images

(The two distributive laws are respectively called the left distributive law and the right distributive law.)

We usually write ab instead of a · b. The identity of the ...

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