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2. Module (mathematics) - Wikipedia

   en.wikipedia.org/wiki/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 group, and scalar ...

3. 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 ...

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. 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 .

6. Graded ring - Wikipedia

   en.wikipedia.org/wiki/Graded_ring

   Example: Given an ideal I in a commutative ring R and an R-module M, the direct sum = / + is a graded module over the associated graded ring / +. A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., ⁠ f ( N ...

7. Glossary of commutative algebra - Wikipedia

   en.wikipedia.org/wiki/Glossary_of_commutative...

   2. The grade grade(M) of a module M over a ring R is grade(Ann M,R), which for a finitely generated module over a Noetherian ring is the smallest n such that Ext n R (M,R) is non-zero. 3. The grade of a module M over a Noetherian local ring with maximal ideal I is the grade of m on I. This is also called the depth of M. This is not consistent ...