Notes for Commutative Algebra M5p55
نویسنده
چکیده
1. Rings and ideals Definition 1.1. A quintuple (A,+, ·, 0, 1) is a commutative ring with identity, if A is a set, equipped with two binary operations; addition + and multiplication ·, and two element 0, 1 ∈ A such that: (1) the triple (A,+, 0) is an abelian group, (2) multiplication is associative: (x · y) · z = x · (y · z), commutative: x · y = y · x and distributive over addition: x · (y + z) = x · y + x · z, (3) we have x · 1 = 1 · x = x. It is rather usual to drop the dot from the notation when we write the product of elements, that is, to write xy instead of x · y. It is also a usual abuse of notation to let just the symbol A denote this whole package.
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