12/13/2023 0 Comments Mathematica modules![]() ![]() 4 Constructions in the theory of modules.ġ) Any Abelian group $M$ is a module over the ring of integers $\ZZ$.Subsequent developments of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf. Krull, it was observed that it was more convenient to formulate and prove many results in terms of arbitrary modules, and not just ideals. At first the theory of modules was developed primarily as a theory of ideals of a ring. Frobenius), led to the study of ideals of certain non-commutative rings. At approximately the same time research on finite-dimensional associative algebras, in particular, group algebras of finite groups (B. Kronecker, devoted to the arithmetic of algebraic number and function fields. The general notion of a module was first encountered in the 1860's till 1880's in the work of R. Gauss as class groups of binary quadratic forms. ![]() The simplest examples of modules (finite Abelian groups they are $\ZZ$-modules) were known already to C.F. Below only left $A$-modules are discussed. When $A$ is commutative, any left $A$-module can be considered as a right $A$-module and the distinction between left and right modules disappears. Any right $A$-module can be considered as a left $A^\text$-modules, and conversely. Right $A$-modules are defined similarly axiom 3) is replaced by $(ma_1)a_2 = m(a_1 a_2)$. A module with this property is called unitary or unital (cf. Ring with identity, then it is usual to require in addition that for any $m \in M$, $1m = m$. An additive Abelian group $M$ is called a left $A$-module if there is a mapping $A\times M \to M$ whose value on a pair $(a, m)$, for $a \in A$, $m \in M$, written $am$, satisfies the axioms:ģ) $a_1(a_2 m) = (a_1 a_2) m$. A module is a generalization of a (linear) Abelian group with the distributive action of a ring.
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