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_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- 32. Consider the set . a. Construct addition and multiplication tables for, using the operations as defined in . b. Observe that is a commutative ring with unity, and compare this unity with the unity in . c. Is a subring of ? If not, give a reason. d. Does have zero divisors? e. Which elements of have multiplicative inverses?arrow_forwarda. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].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_forward
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forward27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.arrow_forwardGiven 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_forward
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