and sufficient conditions for The necessary a non-empty subset S of a ring R to be a subring of R are (i) S+(-S)=S (ii) SSCS.
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- 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.If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.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 unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Exercises If and are two ideals of the ring , prove that is an ideal of .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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 .)[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]