Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and zero mappings.
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- 36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.
- Label each of the following statements as either true or false. The ideals of a ring R and the kernel of the homomorphisms from R to another ring are the same subrings of R.40. Let be idempotent in a ring with unity. Prove is also idempotent.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.