Abstract algebra a10- Let S = {|0 0; a, b are real numbers} be a subring of M(R), and let R×R= {(a, b); a and bin R}. Show that there is no isomorphism between the two rings S and RxR.11- Let R be a ring and let H and K be subrings of R. Let G = HOK. Show that G is subring ofR.12- Show that the ring Z × Z2 is not isomorphic to the ring z.Good Luck

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 18E: 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is...

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18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.
17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
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12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .
Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.
32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.
Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.

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(a b 10- Let S = { ; a, b are real numbers} be a subring of M(R), and let R×R= {(a, b); a and b in R}. Show that there is no isomorphism between the two rings S and R×R.