Can an Element of an Algebraic Structure have Multiple Identities? 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!
 A: The identity element is unique. This can be see from the fact that
$$i = i \star h = h$$
A: $0$ 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.
A: 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.
A: 
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.

