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- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for some
- [Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]14. 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.19. Find a specific example of two elements and in a ring such that and .
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.Suppose 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.If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
- 14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .40. Let be idempotent in a ring with unity. Prove is also idempotent.12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .