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Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
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.
Given right R-module E and left R-module F, there is a canonical homomorphism θ : F ∗ ⊗ R E ∗ → L R (F × E, R) such that θ(f′ ⊗ e′) is the map (f, e) ↦ f, f′ ⋅ e′, e . [citation needed] Thus, an element of a tensor product ξ ∈ F ∗ ⊗ R E ∗ may be thought of giving rise to or acting as an R-bilinear map F × E → R.
A module P is projective if and only if every short exact sequence of modules of the form . is a split exact sequence.That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that f h = id P .
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Module, in connection with modular decomposition of a graph, a kind of generalisation of graph components; Modularity (networks), a benefit function that measures the quality of a division of a Complex network into communities; Protein module or protein domain, a section of a protein with its own distinct conformation, often conserved in evolution
Since a simple module is indecomposable, a semisimple decomposition is an indecomposable decomposition (but not conversely). If the endomorphism ring of a module is local, then, in particular, it cannot have a nontrivial idempotent: the module is indecomposable. Thus, a decomposition with local endomorphism rings is an indecomposable decomposition.