Answer to Question #44360 in Abstract Algebra for Jasvinder Singh
i) R = Q, S = f a
b 2 Q b is not divisible by 3g.
ii) R is the set of all real valued functions on R and S is the set of linear combinations of the
functions fI;cosnt; sinntg where I : R!R is defined by I(x) = 1 for all x 2 R.
ii) Any element of S is a finite trigonometric series, so sum o any pair of elements of S is again in S. Using trigonometric formulas for the products of sin and cos in different combinations one can see that they can be again expressed as the linear combinations of sin and cos of some other arguments. Thus any sum and a product of elements of S again belongs to S, and S becomes a subring of R.
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