- Mark each of the following true or false. a. nZ has zero divisors if n is not prime. b. Every field is an integral domain. c. The characteristic of nZ is n.
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- Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.a. 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 ].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.
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.Let R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.19. Find a specific example of two elements and in a ring such that and .
- 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 .)Prove that a finite ring R with unity and no zero divisors is a division ring.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.