Which of the following statements are true, and which are false? Give reasons for your answers. i) For any ring R and . a, b R, (a b) a 2ab b 2 2 2 ∈ + = + + ii) Every ring has at least two elements. iii) If R is a ring with identity and I is an ideal of R, then the identity of R I is the same as the identity of R. iv) If f : R → S is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +). v) If R is a ring, then any ring homomorphism from R ×R into R is surjective
(i) True: Suppose R is a commutative ring. Then for every
so by the first equation above, .
(ii) False: There is a unique ring with one element called the zero ring. It is a really bad idea to adopt the convention that the zero ring isn't a ring; without it the category of rings fails to have a terminal object and the tensor product of two rings (which is the coproduct for commutative rings) fails to exist in general.
(iii) True:As with subgroups, we call ideal I proper in R if I = 0 and I= R. Noticethat if R has identity (or “unity”) R and if R is in ideal I then I = R. In fact,if u is a unit in R and u ∈ I, then R = u−1 ∈ I and so I = R. If R ∈ thenI = R. So we can say that a left (or right) nonzero ideal I of ring R with unity Ris proper if and only if I contains no units of R. So a division ring (in which everynonzero element is a unit) has no proper left (or right) ideals.
(iv) False:Let ϕ_(R) = s. Observe that
Then we have , and equivalently . Note that s − 1 = 0 by construction.
Now suppose that , then, for any r in R
and thus, is the zero homomorphism, a contradiction. We conclude that
holds for s = 0. Hence R is not surjective.
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