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- 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[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]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 .)
- Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .
- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.