Adv Alegbra HW Solutions 83

Adv Alegbra HW Solutions 83 - 83 (viii) If X is an...

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Unformatted text preview: 83 (viii) If X is an infinite set, then the family of all finite subsets of X forms a subring of the Boolean ring B( X ). Solution. True. 3.2 Prove that a commutative ring R has a unique one 1; that is, if e ∈ R satisfies er = r for all r ∈ R , then e = 1. Solution. Assume that er = r for all r ∈ R . In particular, e1 = 1. On the other hand, the defining property of 1 gives e1 = e. Hence, 1 = e. 3.3 Let R be a commutative ring. (i) Prove the additive cancellation law. Solution. Absent. (ii) Prove that every a ∈ R has a unique additive inverse: if a + b = 0 and a + c = 0, then b = c. Solution. Absent. (iii) If u ∈ R is a unit, prove that its inverse is unique: if ub = 1 and uc = 1, then b = c. Solution. Absent. 3.4 (i) Prove that subtraction in Z is not an associative operation. Solution. In Z, (a − b) − c = a − (b − c) = a − b + c as long as c = 0. (ii) Give an example of a commutative ring R in which subtraction is associative. Solution. If R = F2 , subtraction is the same as addition, and so it is associative. 3.5 Assume that S is a subset of a commutative ring R such that (i) 1 ∈ S ; (ii) if a , b ∈ S , then a + b ∈ S ; (iii) if a , b ∈ S , then ab ∈ S . (In contrast to the definition of subring, we are now assuming a + b ∈ S instead of a − b ∈ S .) Give an example of a commutative ring R containing such a subset S which is not a subring of R . Solution. Let R = Z. If S is the subset consisting of all the positive integers, then S satisfies the new axioms, but it is not a subring because it is not closed under subtraction: for example, S does not contain 1 − 2. 3.6 Find the multiplicative inverses of the nonzero elements in I11 . Solution. Absent. ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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