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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 ...
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 ...
Functions and their modularly separated from each other in the same manner, by the use of function arguments, return values and variable scopes. The main difference between the styles is that functional programming languages remove or at least deemphasize the imperative elements of procedural programming.
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 built-in function, or builtin function, or intrinsic function, is a function for which the compiler generates code at compile time or provides in a way other than for other functions. [23] A built-in function does not need to be defined like other functions since it is built in to the programming language. [24]
With a direct product, we get some natural group homomorphisms for free: the projection maps defined by :, (,) =:, (,) = are called the coordinate functions. Also, every homomorphism f {\displaystyle f} to the direct product is totally determined by its component functions f i = π i ∘ f . {\displaystyle f_{i}=\pi _{i}\circ f.}
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 : is called an R-module homomorphism or an R-linear map if for any x, y in M and r in R,
number of readily identifiable functions and modules within each function and; whether each identifiable function is a manageable entity or should be broken down into smaller components. A structure chart is also used to diagram associated elements that comprise a run stream or thread.