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I'm wondering if an element of an algebraic structure can have two (or more) two-sided identities. Google wasn't very helpful, and I have never encountered anything with the given properties.

Essentially, I'm looking for $g,h,i \in X$ such that $X$ is an algebraic structure, $ig=gi=g, hg=gh=g,$ and $h \neq i$

I have a basic knowledge of algebraic structures, and would appreciate if someone could provide an example of an algebraic structure that contains an element with two or more two-sided identities, or a brief explanation of why it is not possible.

If this property varies based on the type of algebraic structure, the algebraic structures in which I am most interested are Groups, Rings, and Fields.

Thanks for the help!

user2570465
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  • If a given binary operation has a left identity and a right identity, then they are equal. But rings and fields have more than one binary operation. – Zhen Lin Dec 31 '13 at 02:09
  • oops, I forgot to mention that if there are multiple operators, it doesn't matter which one. As examples, can there be a ring that has an element with two different multiplicative identities? can there be a ring that has an element with two different additive identities? please let me know if what I'm saying makes more sense. I may not be using the right terminology. – user2570465 Dec 31 '13 at 02:15

4 Answers4

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The identity element is unique. This can be see from the fact that $$i = i \star h = h$$

MattAllegro
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John Smith
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    I had been thinking that too. But I remember there was the phrase "the element's identity" not "the identity of elements." So, while i and h might be g's identities, can you be sure that h is i's identity and i is h's identity? – user2570465 Dec 31 '13 at 02:35
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$0$ can have multiple 'individual identities' in any ring, w.r.t multiplication:

$x\cdot 0=0\cdot x=0$ for all $x$.

Similarly, e.g. $3=1\cdot 3\equiv 5\cdot 3\pmod{12}$, so $1$ and $5$ and $9$ are identities for $3$ in $\Bbb Z/12\Bbb Z$.

In a group, as we can cancel out, every element must have only one identity. In a semigroup, however, we can play around with individual identities.

Berci
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Can an element of an algebraic structure have multiple identities?

If any magma (or algebraic structure on a set equipped with one binary operation) has one identity element, than it is unique. Instead, left and right identity elements may not be unique. More precisely:

  • if one identity element exists, than no other element can be a left identity nor a right identity;
  • if two or more left (right) identities exist, than no element can be a right (left) identity.
MattAllegro
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Hint $\ $ If $\ fg = 0\ $ then $\ (f\!+\!1)g = g = 1\cdot g,\ $ e.g. in the ring $\ \Bbb Z/fg = $ integers mod $\,fg,\,\ f\ne 0$.

For example $\ f,g = 4,3\,$ yields the example in Berci's answer.

Bill Dubuque
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