n. a subring of a ring that is closed under subtraction and under multiplication by any ring element whatever. In the non-commutative case, one distinguishes left ideals and right ideals. In the absence of qualification as a left or right ideal, the term may be taken to refer to a two-sided ideal, that is, one closed under multiplication both on the left and on the right; this is occasionally called a normal subring. The multiples of any fixed integer form an ideal in the ring of integers. An ideal in a ring R is an R-module.