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If A and B are sets and every element of A is also an element of B, then: . A is a subset of B, denoted by , or equivalently,; B is a superset of A, denoted by .; If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then:
Ideals appear naturally in the study of modules, especially in the form of a radical. For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings. Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module.
A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then RE is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E.
S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all.
If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for proper subsets. For clarity, one can ...
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
proper If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is reflexive), then the qualification proper requires the objects to be different. For example, a proper subset of a set S is a subset of S that is different from S, and a proper divisor of a number n is a divisor of n that is ...
A number of conventions have been developed to make Wikipedia's mathematics articles more consistent with each other. These conventions cover choices of terminology, such as the definitions of compact and ring, as well as notation, such as the correct symbols to use for a subset.