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In a noncommutative ring R, a right ideal is a subset I which is an additive subgroup of R and such that for all r in R and all a in I, ar in I. (1) For all a in R, the set ...

A group action G×X->X is effective if there are no trivial actions. In particular, this means that there is no element of the group (besides the identity element) which does ...

Two groups G and H are said to be isoclinic if there are isomorphisms G/Z(G)->H/Z(H) and G^'->H^', where Z(G) is the group center of the group, which identify the two ...

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 ...

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. ...

Suppose that E(G) (the commuting product of all components of G) is simple and G contains a semisimple group involution. Then there is some semisimple group involution x such ...

Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms Delta:H->H tensor _RH (1) (comultiplication) and ...

The underlying set of the fundamental group of X is the set of based homotopy classes from the circle to X, denoted [S^1,X]. For general spaces X and Y, there is no natural ...

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

An R-module M is said to be unital if R is a commutative ring with multiplicative identity 1=1_R and if 1m=m for all elements m in M.