Let R be a commutative ring with unity and c E R. The map R[X] → R[X] f(X) → R= f(X + c) is an isomorphism of rings.
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- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of24. 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 .)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 ].
- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .Label each of the following statements as either true or false. Every quotient ring of a ring R is a homomorphic image of R.