If Sii I is a family of subrings of a commutative ring R pro

If (S_i)i I is a family of subrings of a commutative ring R prove that intersection is also a subring of R. If X is a subset of a commutative ring R, define G(X), the subring generated by X, to be the intersection of all the subrings of R that contain X Let (S, ) be a family of subrings of a commutative ring R, each of which is a field. Prove that the subring Conclude that the into section of a family of subfields of a field is a subfield. Let p be a prime and let A_p be the set of all fractions with denominator a power of p. Show, with the usual operations of addition and multiplication, that A_p is a subring of Q. Describe the smallest subring of Q that contains both A_2 and A_5. Let p be a prime and let Q_p be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by p.

Solution

4.62

Since p is prime any number of the form a/p is in its lowest form.

A_p has elements of the form a/p^k for k in whole numbers

The product would be a/p^k * b/p^l = ab/p^(k+l) belongs to A_p

Additive identity is 0/p^k = 0

Distributive and inverse is negative

Hence group under addition

Also commutative

Hence a subring under addition.

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A₂=a/2^k and A₅= a/5^k

The two rings are disjoint.

Hence smallest ring of Q that contains both = {x: x is in A₂ or x is in A₅}