I was thinking about two-sided ideals and have some intuition-guided, soft questions regarding them. Since I don't have anyone to talk to about such subject matter, I thought I'd ask.
Let $a$ be an element of $A$, a two-sided ideal of the ring $R$. If $R$ is not commutative, then the possibility arises where $ra \neq ar$, $r$ being an element of $R$ (as in the case of a matrix ring over a ring $R$ and its two-sided, non-commutative ideal $M_n(A)$).
Then $A$ contains the same elements in both its left and right, but the structure relating those elements must differ.
Is there a classification of two-sided ideals that are symmetrical in right and left? If a two-sided ideal has asymmetry, are there conditions that cause it to be simple (like the matrix ring with elements from a field)?