Let R be a commutative ring and M₂(R) the ring of 2-by-2 matrices with entries in R (with the usual matrix addition and multiplication). Prove that if I is an ideal in R then M2(I) is an ideal in M₂(R).
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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Exercises If and are two ideals of the ring , prove that is an ideal of .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.
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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.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 .)
- 44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.15. Let and be elements of a ring. Prove that the equation has a unique solution.Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .
- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.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 ].19. Find a specific example of two elements and in a ring such that and .