two-sided ideal

In a commutative ring an ideal is just a set you can multiply into and stay inside. In a noncommutative ring multiplication has two sides, so you must ask whether the set absorbs products from the left, from the right, or from both. A two-sided ideal is the demanding one: it absorbs multiplication coming from either side. It is precisely the kind of subset you can quotient by and still get a ring, which is why it is the central notion for forming quotients.

Precisely, a two-sided ideal of a ring R is an additive subgroup I such that r*x is in I and x*r is in I for every r in R and x in I. Left ideals require only the first condition, right ideals only the second; a two-sided ideal is exactly a subgroup that is simultaneously a left and a right ideal. The quotient R/I inherits a well-defined ring structure precisely because I is two-sided.

Two-sided ideals are the kernels of ring homomorphisms: a subset is the kernel of some homomorphism out of R if and only if it is a two-sided ideal, and the isomorphism theorems then run exactly as in the commutative case. This is what makes the notion indispensable; one-sided ideals, while important for module theory, are not kernels and do not give quotient rings.

The supply of two-sided ideals measures how far a ring is from simple. A ring is simple exactly when its only two-sided ideals are 0 and itself. Strikingly, a ring can be teeming with one-sided ideals while having almost no two-sided ones: M_n(k) over a field has many left and right ideals but is simple, with no two-sided ideals besides the trivial pair.