Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardIf R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.arrow_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_forward22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)arrow_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward
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