MATH 436 Notes: Rings
نویسنده
چکیده
1 Rings Definition 1.1 (Rings). A ring R is a set with two binary operations + and · called addition and multiplication respectively such that: (1) (R, +) is an Abelian group with identity element 0. (2) (R, ·) is a monoid with identity element 1. (3) Multiplication distributes over addition: a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a for all a, b, c ∈ R. A ring is called commutative when (R, ·) is commutative. On occasion we will consider rings that satisfy a weaker version of (2) where we drop one or more of the conditions of a monoid. If (R, ·) is a semi-group without 1 we say R is a ring without identity and if (R, ·) is nonasso-ciative we call R a nonassociative ring. A motivating example of a ring is the integers Z under the usual addition and multiplication. In general we try to mimic basic properties of this ring in the study of other rings. ·) is a ring and S ⊆ R is itself a ring under the same operations (with the same multiplicative identity 1) then we call S a subring of R. For example, the integers Z are a subring of the rational numbers Q.
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