We will assume rings have identity and ring homomorphisms are unital (send 1 to 1) unless stated otherwise. Do Problems A-C but do not turn these in. Turn in Problems 1-9. \text { Let } R \text { be a ring. Show that }(-1)^{2}=1 \text { in } R \text {. } Problem B. Decide which of the following are subrings of the ring of all functions from the closed interval [0, 1] to R: (a) the set of all functions f(r) such that f(q) = 0 for all q € Qn [0, 1] (b) the set of all polynomial functions (c) the set of all functions which only have a finite number of zeros, together with the zero function (d) the set of all functions which have an infinite number of zeros (e) the set of all functions f such that lim ₁- f(x) = 0.1- (f) the set of all rational linear combinations of the functions sin(nx) and cos(mr), wherem. n € (0.1.2. } . Decide which of the following are ideals of the ring Z x Z: \text { (a) }\{(a, a) \mid a \in \mathbb{Z}\} \text { (b) }\{(2 a, 2 b) \mid a, b \in \mathbb{Z}\} \text { (c) }\{(2 a, 0) \mid a \in \mathbb{Z}\} \text { (d) }\{(a,-a) \mid a \in \mathbb{Z}\} \text {. }

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