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1. Let R be a commutative ring with unity a. Let ✅ be a collection of ideals in R. Prove that ea is an ideal in R. b. Let X be a subset of R. Prove that there exists a smallest ideal in R which contains X. (An ideal a is said to be smallest ideal satisfying a given condition c, if for every ideal b satisfying condition c, a Cb. That is, a Cb whenever b is an ideal satisfying condition c.) This ideal is denoted by < X >. c. If a € R, the smallest ideal in R containing a is the given by {ra : r € R}. (We denote this ideal by .)

1. Let R be a commutative ring with unity a. Let ✅ be a collection of ideals in R. Prove that ea is an ideal in R. b. Let X be a subset of R. Prove that there exists a smallest ideal in R which contains X. (An ideal a is said to be smallest ideal satisfying a given condition c, if for every ideal b satisfying condition c, a Cb. That is, a Cb whenever b is an ideal satisfying condition c.) This ideal is denoted by < X >. c. If a € R, the smallest ideal in R containing a is the given by {ra : r € R}. (We denote this ideal by .)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 31E: Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set...
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1. Let R be a commutative ring with unity a. Let ✅ be a collection of ideals in R. Prove that ea is an ideal in R. b. Let X be a subset of R. Prove that there exists a smallest ideal in R which contains X. (An ideal a is said to be smallest ideal satisfying a given condition c, if for every ideal b satisfying condition c, a Cb. That is, a Cb whenever b is an ideal satisfying condition c.) This ideal is denoted by < X >. c. If a € R, the smallest ideal in R containing a is the given by {ra : r € R}. (We denote this ideal by <a >.)
Transcribed Image Text:1. Let R be a commutative ring with unity a. Let ✅ be a collection of ideals in R. Prove that ea is an ideal in R. b. Let X be a subset of R. Prove that there exists a smallest ideal in R which contains X. (An ideal a is said to be smallest ideal satisfying a given condition c, if for every ideal b satisfying condition c, a Cb. That is, a Cb whenever b is an ideal satisfying condition c.) This ideal is denoted by < X >. c. If a € R, the smallest ideal in R containing a is the given by {ra : r € R}. (We denote this ideal by <a >.)
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