How many of the following statements are false?
a) Subring of a ring is a ring.
b) Subring of commutative ring is a commutative ring.
c) Subring of a integral domain is an integral domain.
d) Subring of a field is a field.
closed as not constructive by Andres Caicedo, Steve D, William, rschwieb, LVK Sep 7 '12 at 21:33
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To answer these questions, carefully write down the properties of a ring/comm. ring/int. domain, and those of a subring, and then check whether or not all the properties are satisfied by the subring (or if the statement is false, find a counterexample). If you need intuition, the integers are an integral domain (and so a comm. ring, and so a ring); see whether you believe these statements are true for the integers.
For instance, for a), let R be the ring, S the subring. Because R is a ring, we have $$\forall a, b \in R, a + b \in R$$ $$\forall a, b, c \in R, (a + b) + c = a + (b + c)$$ $$\exists 0 \in R, : \forall a \in R, 0 + a = a + 0 = a$$ $$\forall a \in R, \exists b (= "-a") \in R : a + b = b + a = 0$$ $$\forall a, b \in R, a + b = b + a$$ $$\forall a, b \in R, a · b \in R$$ $$\forall a, b, c \in R, (a · b) · c = a · (b · c)$$ $$\exists 1 \in R : \forall a \in R, 1 · a = a · 1 = a$$ $$\forall a, b, c \in R, a · (b + c) = (a · b) + (a · c)$$ $$\forall a, b, c \in R, (a + b) · c = (a · c) + (b · c)$$ Because $S$ is a subring, $$\forall a, b \in S, a + b \in S$$ $$\forall a, b \in S, a · b \in S$$ $$\forall a \in S, −a \in S$$ $$1 \in S$$ Now, verify that $S$ satisfies the remaining properties of a ring (it does). It's a bit tedious, but that's how it's done. (if the text you're working from doesn't require rings to be unital, you can ignore the properties involving $1$.)
For the subsequent parts, you can build on the earlier parts; for instance, to show (b), you can start by claiming the subring is (at least) a ring by part (a), and it just remains to show it is commutative.