Does anyone know how to do this question?
In each of the following cases, decide whether or not the two rings are isomorphic.
(a) Z and Q.
(b) C and H.
(c) Z6 × R and R × Z6.
(d) Q and Q × Z2.
(e) R and C.
Thanks very much.
(a) No. The integers are not dense. And the rationals are dense.Originally Posted by suedenation
(b)No. Because is a commutative ring, while the quaternion numbers are not. Because for example, while
(c)Yes. I really do not see how order is important in a direct product between rings.
(e)No. The reals are ordered but the complex numbers cannot be ordered.
I am thinking about (d)
(d) Nope.
Explanation
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It is known that the direct product between two groups is a group. And also the direct product between two rings is a ring. But the direct product between two fields IS NOT a field EVER.
Proof: Let and be fields. Then, by definition a field is a commutative division ring. Meaning that every non-zero element in a field has a multiplicative inverse. Thus, we have . Now, acts like a zero. Thus, we need to show that for all has an inverse where not both are zero. BUT is a nonzero element where has no multiplicative inverse. Because thus, there is no way how it can produce which acts like a multiplicative inverse in
Thus, is a field. And is also a field. But cannot be field. Thus, there is no isomorphism between and because it is not a field.
Note: By I mean that which is the additive identity and which is the additive identity. Similary .
Q.E.D.