(a) Let R be a ring, a € R and define AR = {r € R|ar = ar = OR} where OR ER is the zero element of R. Then check as to whether AR is a subring of R or not
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- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- Prove that a finite ring R with unity and no zero divisors is a division ring.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]