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- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.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 .)22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.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+y4Let 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.
- 12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .