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Elements Of Modern Algebra
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for somearrow_forward
- 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)arrow_forward14. Let be an ideal in a ring with unity . Prove that if then .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_forward
- Let 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_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,