Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 12, Problem 12E
Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides c if and only if abdivides c.
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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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- 19. Find a specific example of two elements and in a ring such that and .arrow_forward12. Let be a commutative ring with unity. If prove that is an ideal of.arrow_forwardSuppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a multiplicative inverse, then b=c.arrow_forward
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.arrow_forward
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forwardLet a0 in the ring of integers . Find b such that ab but (a)=(b).arrow_forward
- Given that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forwardLet R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s, (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. Prove that RS is commutative if both R and S are commutative. Prove RS has a unity element if both R and S have unity elements. Given as example of rings R and S such that RS does not have a unity element.arrow_forwardAn element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().arrow_forward
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