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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.[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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}.A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.
- 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.)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 ].Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4