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2. Module (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Module_(mathematics)

   Module (mathematics) In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module also generalizes the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers . Like a vector space, a module is an additive abelian ...

3. Graded ring - Wikipedia

   en.wikipedia.org/wiki/Graded_ring

   t. e. In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that ⁠ ⁠. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid.

4. Depth (ring theory) - Wikipedia

   en.wikipedia.org/wiki/Depth_(ring_theory)

   Depth (ring theory) In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the ...

5. Linear algebra - Wikipedia

   en.wikipedia.org/wiki/Linear_algebra

   One may thus replace the field of scalars by a ring R, and this gives the structure called a module over R, or R-module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms ) are defined for modules exactly as for vector spaces, with the essential difference that, if R is not a field, there are ...

6. Hilbert's syzygy theorem - Wikipedia

   en.wikipedia.org/wiki/Hilbert's_syzygy_theorem

   Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring in n indeterminates over a field k, then the n th syzygy module of M is always a free module . In modern language, this implies that the projective dimension of M is at most n, and thus that there exists a free resolution. of length k ≤ n .

7. Modular arithmetic - Wikipedia

   en.wikipedia.org/wiki/Modular_arithmetic

   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. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...