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The Cartesian product of a finite or infinite set of modules over a ring with only finitely many nonzero entries in each sequence.

The group direct sum of a sequence {G_n}_(n=0)^infty of groups G_n is the set of all sequences {g_n}_(n=0)^infty, where each g_n is an element of G_n, and g_n is equal to the ...

A module M over a unit ring R is called flat iff the tensor product functor - tensor _RM (or, equivalently, the tensor product functor M tensor _R-) is an exact functor. For ...

The direct limit, also called a colimit, of a family of R-modules is the dual notion of an inverse limit and is characterized by the following mapping property. For a ...

A tensor having contravariant and covariant indices.

Although the multiplication of one vector by another is not uniquely defined (cf. scalar multiplication, which is multiplication of a vector by a scalar), several types of ...

A Lorentz tensor is any quantity which transforms like a tensor under the homogeneous Lorentz transformation.

The standard Lorentzian inner product on R^4 is given by -dx_0^2+dx_1^2+dx_2^2+dx_3^2, (1) i.e., for vectors v=(v_0,v_1,v_2,v_3) and w=(w_0,w_1,w_2,w_3), ...

If (f,U) and (g,V) are functions elements, then (g,V) is a direct analytic continuation of (f,U) if U intersection V!=emptyset and f and g are equal on U intersection V.

A tensor category (C, tensor ,I,a,r,l) is strict if the maps a, l, and r are always identities. A related notion is that of a tensor R-category.