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A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
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.
A modular function is a function that is invariant with respect to the modular group, but without the condition that f (z) be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic: they are holomorphic on the complement of a set of isolated points, which are poles of the function.
A Drinfeld module is a module over a ring of functions on algebraic curve with ... is the difference ... An isomorphism between modules is an invertible ...
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring R , then a function f : M → N {\displaystyle f:M\to N} is called an R - module homomorphism or an R - linear map if for any x , y in M and r in R ,
With modular programming, concerns are separated such that modules perform logically discrete functions, interacting through well-defined interfaces. Often modules form a directed acyclic graph (DAG); in this case a cyclic dependency between modules is seen as indicating that these should be a single module. In the case where modules do form a ...
At the same time, the mapping of a function to the value of the function at a point is a functional; here, is a parameter. Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals .