The necessary and sufficient conditions for a non-empty subset S of a ring R to be a subring of R are (i) S+(-S)=S (ii) SS⊆S.
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The necessary and sufficient conditions for a non-empty
subset S of a ring R to be a subring of R are
(i) S+(-S)=S (ii) SS⊆S.
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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.15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .True or False Label each of the following statements as either true or false. 4. If a ring has characteristic zero, then must have an infinite number of elements.
- Label each of the following statements as either true or false. The ideals of a ring R and the kernel of the homomorphisms from R to another ring are the same subrings of R.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal ofTrue or False Label each of the following statements as either true or false. 3. The characteristic of a ring is zero if is the only integer such that for all in.