a. Give an example where
Give an example where
Prove that the set of all elements in a ring
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Chapter 5 Solutions
Elements Of Modern Algebra
- 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 .arrow_forwardProve that a finite ring R with unity and no zero divisors is a division ring.arrow_forwardAn 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.arrow_forward
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forward
- 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}.arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forwardLet 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+y4arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,