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 .)
- 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 .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.
- 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 ].17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.19. Find a specific example of two elements and in a ring such that and .