College Algebra Exam Review 268

College Algebra Exam Review 268 - 278 6. RINGS Definition...

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Unformatted text preview: 278 6. RINGS Definition 6.2.24. The smallest left ideal containing a subset S is called the left ideal generated by S . The smallest left ideal containing a single element x 2 R is called the principal left ideal generated by x . When R has an identity element the principal left ideal generated by x is just Rx D frx W r 2 Rg: See Exercise 6.2.8 Proposition 6.2.23 and Definition 6.2.24 have evident analogues for right ideals. The following is the analogue for two-sided ideals: Proposition 6.2.25. Let R be a ring and S a subset of R. Let hS i denote the additive subgroup of R generated by S . (a) Define RS R D fa1 s1 b1 C a2 s2 b2 C (b) (c) C an sn bn W n 2 N ; an ; bn 2 Rg: Then RS R is a two-sided ideal. hS i C RS C S R C RS R is the smallest ideal of R containing S , and is equal to the interesection of all ideals of R containing S If R has an identity element , then hS i C RS R D RS R. Proof. Essentially the same as the proof of Proposition 6.2.23. I Definition 6.2.26. The smallest ideal containing a subset S is called the ideal generated by S , and is denoted by .S /. The smallest ideal containing a single element x 2 R is called the principal ideal generated by x and is denoted by .x/. When R has an identity element, the principal ideal generated by x 2 R is .x/ D fa1 xb1 C a2 xb2 C C an xbn W n 2 N ; ai ; bi 2 Rg: See Exercise 6.2.9. When R is commutative with identity, ideals and left ideals coincide, so .x/ D Rx D frx W r 2 Rg: The ideal generated by S is, in general, larger than the subring generated by S ; for example, the subring generated by the identity element consists of integer multiples of the identity, but the ideal generated by the identity element is all of R. ...
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