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. [1]

Apr 12, 2025 · Every module over a ring R contains a so-called "zero element" which fulfils the properties suggested by its name with respect to addition, 0+0=0, and with respect to multiplication by any element a of R, a·0=0.

In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus.

Oct 20, 2023 · Modulo calculator finds a mod b, the remainder when a is divided by b. The modulo operation returns the remainder in division of 2 positive or negative numbers or decimals.

Definitions and basic properties. 1.1. Left and right modules, vector spaces, and algebras. 1.2. Examples and constructions of modules. 1.3. Elementary properties of modules. 1.4. Submodules. 1.5. Generating sets of modules. 1.6. Quotient modules. 1.7. Torsion elements of a module and the torsion submodule. 1.8. Annihilators. 1.9.

2 days ago · math. isclose (a, b, *, rel_tol = 1e-09, abs_tol = 0.0) ¶ Return True if the values a and b are close to each other and False otherwise. Whether or not two values are considered close is determined according to given absolute and relative tolerances.

May 24, 2024 · Modular arithmetic, also known as clock arithmetic, deals with finding the remainder when one number is divided by another number. It involves taking the modulus (in short, ‘mod’) of the number used for division. If ‘A’ and ‘B’ are two integers such that ‘A’ is divided by ‘B,’ then: $ {\dfrac {A} {B}=Q,remainderR}$ Here,

A free module is isomorphic to $R^n$, but is $n=0$ allowed? Alternatively, a free module is defined to have a set of linearly independent generating sets, in this case it would be $0$ but I recall by definition in the case of vector spaces that $0$, by itself, is dependent.

A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. rx ≠ 0 for some x in M ). Equivalently, the annihilator of M is the zero ideal.

Module (absolute value) of a positive number or zero is the number itself and module of a negative number is called its contrary number i.e. From the definition is clear that the absolute value of every rational number different from zero is a positive number. Therefore contrary numbers have equal modules.