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Math 361—Classification of Rings Classify each of the following structures according to the chart below. Apply the strongest label possible. The symbols + and · indicate ordinary addition and multiplication. If the structure is not a ring, identify at least one ring property that fails. If the structure is a ring, but not a field, identify at least one property that fails. Additionally, identify any units in a ring with unity. The variables n and m refer to positive integers. N,Z,Q,R, and C refer, respectively, to the sets of Natural Numbers, Integers, Rational Numbers, Real Numbers, and Complex Numbers. Structure ({0}, +, ·) (Z,+, ·) (N, +, ·) (nZ, +, ·) (Zn , +, ·) (tricky!) (R, +, ·) (R, +, ÷) (C,+,·) (Q,+,·) (Mn (R), +, ·) (Mn,m (R), +, ·) (Q, +, * ) where a*b = ab/2 (Q[x], +, ·) (R[x], +, ·) (C[x], +, ·) Z[√2] = {a + b√2 | a, b ∈ Z} with the usual +, · Co(R) ≡ the set of continuous functions defined over the real line. C1(R) ≡ the set of once differentiable functions defined over the real line. Co((a,b)) ≡ the set of continuous functions defined over the open interval (a,b). C1((a,b)) ≡ the set of once differentiable functions defined over (a,b). C2((a,b)) ≡ the set of twice differentiable functions defined over (a,b). Ring? Ring with Comm. Unity? Ring? Integral Units? Domain? Field? What Fails? Math 361—Classification of Rings Part 2. Group Exercise. Write the names of as many rings in the best possible position of the Venn Diagram below. The best possible position is the place where a particular ring enjoys the most properties. Ring Commutative Ring Integral Domain PID Field