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A: The detailed solution is given below. Thank you
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A: As per bartleby guidelines we only give first question answers.
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Q: Q25: What's mean by a zero divisor element of a ring R? Find all zero divisor elements of Zg-
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- a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.22. Let be a ring with finite number of elements. Show that the characteristic of divides .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
- An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]
- 7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.Suppose R is a ring in which all elements x are idempotent-that is, all x satisfy x2=x. (Such a ring is called a Boolean ring.) a. Prove that x=x for each xR. (Hint: Consider (x+x)2.) Prove that R is commutative. (Hint: Consider (x+y)2.)