How many ring homomorphisms from field to field?

Asked Apr 08 '18 at 07:51

Active Apr 08 '18 at 08:01

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Abstract algebra related problem Let R be a non zero ring with identity such that $a^ 2 =a$, for all $a\in R$. Then which of the following statements are true.

$(1)$ there is no such ring.
$(2)$ $2a=0$ for all $a\in R$
$(3)$ $3a=0$ for all $a\in R$
$(4)$ $Z/2Z$ is subring of $R$

edited Apr 08 '18 at 08:01

Peter

86,576

asked Apr 08 '18 at 07:51

Poonam Bai

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We can rule out $(1)$ because $\mathbb Z_2$ has the desired property. – Peter Apr 08 '18 at 07:54
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See this question. So $R$ has characteristic $2$ and is commutative. – Dietrich Burde Apr 08 '18 at 07:56
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2 is true: $(a+1)^2 = a^2+2a+1=3a+1$ on the one hand, but on the other hand it is equal to $a+1$, so $2a = 0$. – Patrick Stevens Apr 08 '18 at 07:56
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By the way, the title of this question has nothing to do with its contents, as far as I can see. – Patrick Stevens Apr 08 '18 at 07:58
@PatrickStevens I agree – Peter Apr 08 '18 at 07:59
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$(3)$ must be wrong because $2a=0$ and $3a=0$ would imply $a=0$, but $a$ has an identity element. – Peter Apr 08 '18 at 08:03
@DietrichBurde Naturally. We posted those statements at the same time. – Patrick Stevens Apr 08 '18 at 10:03

How many ring homomorphisms from field to field?

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