Let R be a commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and M, = {x E M; rx 0}. Show that rM and M, are submodules a М.
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- 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 .)Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .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.
- 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.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .