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The direct sum of modules A and B is the module A direct sum B={a direct sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...

Let a module M in an integral domain D_1 for R(sqrt(D)) be expressed using a two-element basis as M=[xi_1,xi_2], where xi_1 and xi_2 are in D_1. Then the different of the ...

The length of all composition series of a module M. According to the Jordan-Hölder theorem for modules, if M has any composition series, then all such series are equivalent. ...

Module multiplicity is a number associated with every nonzero finitely generated graded module M over a graded ring R for which the Hilbert series is defined. If dim(M)=d, ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

In algebraic geometry classification problems, an algebraic variety (or other appropriate space in other parts of geometry) whose points correspond to the equivalence classes ...

Let A={a_1,a_2,...} be a free Abelian semigroup, where a_1 is the identity element, and let mu(n) be the Möbius function. Define mu(a_n) on the elements of the semigroup ...

A one-sided surface reminiscent of the Möbius strip, attributed to Gourmalin (Bouvier and George 1979, p. 477; Boas 1995). This surface is topologically equivalent to a Klein ...

The theorem of Möbius tetrads, also simply called Möbius's theorem by Baker (1925, p. 18), may be stated as follows. Let P_1, P_2, P_3, and P_4 be four arbitrary points in a ...

Möbius tetrahedra, also called Möbius tetrads (Baker 1922, pp. 61-62) are a pair of tetrahedra, each of which has all the vertices lying on the faces of the other: in other ...