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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 8 Homework 8 1. Determine which of the following are rings, with the usual operations of addition and multiplication. If it is a ring you do not need to write out all the axioms, but if it is not a ring, explain why. If the set is a ring, is it a field? If not, explain why. ? ? (a) Qp 2q “ ta ` b 2 ∶ a, b P Qu (b) Zris “ ta ` bi ∶ a, b P Zu, where i “ ? ´1, so that i2 “ ´1 (c) The set of following matrices with entries in the ring Z2 : “ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙* 0 0 1 0 1 1 0 1 , , , 0 0 0 1 1 0 1 1 (d) The set of continuous functions f ∶ R Ñ R. Addition and multiplication is defined pointwise. That is, for two such functions f, g, we have f ` g and f g defined by pf ` gqpxq “ f pxq ` gpxq and pf gqpxq “ f pxqgpxq for x P R. 2. For each ring R given, determine the set of units Rˆ . (a) The integers Z

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