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In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring.The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.
Note that C99 and C++ do not implement complex numbers in a code-compatible way – the latter instead provides the class std:: complex. All operations on complex numbers are defined in the <complex.h> header. As with the real-valued functions, an f or l suffix denotes the float complex or long double complex variant of the function.
The function: / sending a in A to its equivalence class a + B is called the quotient map or the projection map, and is a module homomorphism. The addition operation on A / B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and scalar multiplication of elements of A / B by ...
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.
In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a ...
Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.These are the cyclic groups of prime order.. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules.
Alternatively, the Yoneda product can be defined in terms of resolutions. (This is close to the definition of the derived category.) For example, let R be a ring, with R-modules A, B, C, and let P, Q, and T be projective resolutions of A, B, C. Then Ext i R (A,B) can be identified with the group of chain homotopy classes of chain maps P → Q[i ...