16.1.14. Let R = M2×2 (R) be the ring of two-by-two matrices with real entries. [0 b [a b] Let S = { | a, b = R} and T = { 0 a 00 [o of] b € R} . (a) Are T and S subrings of R? (b) Is T an ideal of S? Is T an ideal of R? Is S an ideal of R?

Consider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.

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.

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .