If E is a ring of even integers, then show that an ideal (4), generated by 4 is a maximal ideal of E.
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- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )
- Let 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.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of22. Let be a ring with finite number of elements. Show that the characteristic of divides .
- 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 .)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.