If R is a ring, then prove that (R, A, n) is a commutative ring in the algebraic sense.
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- Suppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a multiplicative inverse, then b=c.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- 12. Let be a commutative ring with unity. If prove that is an ideal of.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.Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)