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# Using real quaternions as a model, we define the quaternions over the integers mod p, p an odd prime number, in exactly the same way; however, now

Using real quaternions as a model, we define the quaternions over the integers mod p, p an odd prime number, in exactly the same way;

however, now considering all symbols of the form

cx0 + cx1i + cx2j + cx3k, where cx0, cx1, cx2, cx3 are integers mod p.

(a) Prove that this is a ring with p4 elements whose only ideals are (0) and the ring itself.

(b) Prove that this ring is not a division ring.

If R is any ring a subset L of R is called a left-ideal of R if

1.) L is a subgroup of R under addition.

2.) r E R, aEL implies ra E L.

(One can similarly define a right-ideal.) An ideal is thus simultaneously a left- and right-ideal of R.