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Elements Of Modern Algebra
- 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_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forward
- 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+y4arrow_forward14. Let be an ideal in a ring with unity . Prove that if then .arrow_forward32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,