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(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 ).
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.
(ii) Prove that every a ∈ R has a unique additive inverse: if a + b = 0
and a + c = 0, then b = c.
(iii) If u ∈ R is a unit, prove that its inverse is unique: if ub = 1 and
uc = 1, then b = c.
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
Solution. If R = F2 , subtraction is the same as addition, and so it
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.
- Fall '11