{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Adv Alegbra HW Solutions 83

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
83 (viii) If X is an in fi nite set, then the family of all fi nite 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 satis fi es er = r for all r R , then e = 1. Solution. Assume that er = r for all r R . In particular, e 1 = 1. On the other hand, the de fi ning property of 1 gives e 1 = 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 = F 2 , subtraction is the same as addition, and so it
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online