"; that is, is a left ideal if it satisfies the following two conditions: is a subgroup of For every and every , the product is in . A right ideal is defined with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of R is an R-submodule of R when R is viewed as a left (resp. right, bi-) R-module. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring ℤ/nℤ of integers modulo n given integer (note that ℤ is a commutative ring). The key observation here is that we obtain ℤ/nℤ by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy 2 requirements: 1) n must be identified with 0 since n is congruent to 0 modulo n. 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap ℤ around itself). The notion of an ideal arises when we ask the question: What is the exact set of integers that we are forced to identify with 0? The answer is, unsurprisingly, the set of all integers congruent to 0 modulo n. That is, we must wrap ℤ around itself infinitely many times so that the integers ..., , , , , ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤ/nℤ is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that nℤ is an ideal of ℤ. Remark. Identifications with elements other than 0 also need to be made. For example, the elements in must be identified with 1, the elements in must be identified with 2, and so on. Those, however, are uniquely determined by nℤ since ℤ is an additive group. We can make a similar construction in any commutative ring R: start