Does a ring without an identity element even have a characteristic?
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No, it does not. – superAnnoyingUser Dec 06 '13 at 20:31
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1Is there a source that you can point me to that would have more information? Or can you elaborate? – Jenny Dec 06 '13 at 20:32
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I don't think anyone would publish this. We define characteristic for rings with unity. – superAnnoyingUser Dec 06 '13 at 20:38
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@Ivan: the third paragraph of the Wikipedia article you link to explains how the definition can be generalized to non-unital rings in a natural way. – hmakholm left over Monica Dec 06 '13 at 20:43
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One may define the characteristic of an abelian group $A$ to be the non-negative generator of the ideal $\mathrm{Ann}(A)$. Thus, we have $\mathrm{char}(A) \cdot A = 0$, and every other integer with this property is a multiple of $\mathrm{char}(A)$. This coincides with the usual definition of the characteristic of a ring when we consider its underlying additive group. As you see, we don't really need a unit, in fact no multiplication.
Martin Brandenburg
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4I expand the explanation a little more, for people who, like me, may need it: The abelian group A can be seen as a Z-module. Then Ann(A) is the biggest subset of Z that completely annihilates A. It can be checked that A is an ideal of Z, and thus, since Z is a PID, Ann(A)=nZ for some integer n. We define char(A)=n. – Jose Brox Oct 04 '14 at 15:55