Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 12, Problem 9E
Prove that the intersection of any collection of subrings of a ring Ris a subring of R.
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Contemporary Abstract Algebra
Ch. 12 - The ring {0, 2, 4, 6, 8} under addition and...Ch. 12 - Find an integer n that shows that the rings Zn...Ch. 12 - Show that a ring is commutative if it has the...Ch. 12 - Prove that the intersection of any collection of...Ch. 12 - Let a, b, and c be elements of a commutative ring,...Ch. 12 - Let a andb belong to a ring R and let mbe an...Ch. 12 - Show that if m and n are integers and a and b are...Ch. 12 - Show that a ring that is cyclic under addition is...Ch. 12 - Let R be a ring. The center of R is the set...Ch. 12 - Let R be a commutative ring with unity and let...
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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.arrow_forward7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.arrow_forward[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]arrow_forward
- 15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.arrow_forward
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