In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.
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In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.donot copy the answers
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- Let a0 in the ring of integers . Find b such that ab but (a)=(b).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.Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.
- 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.Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
- Exercises 2. Decide whether each of the following sets is a ring with respect to the usual operations of addition and multiplication. If it is not a ring, state at least one condition in Definition 5.1a that fails to hold. The set of all integers that are multiples of . The set of all real numbers of the form with and . The set of all real numbers of the form , where and are rational numbers. The set of all real numbers of the form , where and are rational numbers. The set of all positive real numbers. The set of all complex numbers of the form , where (This set is known as the Gaussian integers.) The set of all real numbers of the form with and . The set of all real numbers of the form with 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 .)52. (See Exercise 51.) a. Write out the elements of and construct addition and multiplication tables for this ring. (Suggestion: Write for, for in.) b. Is a commutative ring? c. Identify the unity elements, if one exists. d. Find all units, if any exist. e. Find all zero divisors, if any exist. f. Find all idempotent elements, if any exist. g. Find all nilpotent elements, if any exist. Exercise 51. 51. Let and be arbitrary rings. In the Cartesian product of and, define if and only if and , , . Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of and and is denoted by. Prove that is commutative if both and are commutative. Prove has a unity element if both and have unity elements. Given as example of rings and such that does not have a unity element.