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Let (K,L) be a pair consisting of finite, connected CW-complexes where L is a subcomplex of K. Define the associated chain complex C(K,L) group-wise for each p by setting ...

A class number formula is a finite series giving exactly the class number of a ring. For a ring of quadratic integers, the class number is denoted h(d), where d is the ...

A filtration of ideals of a commutative unit ring R is a sequence of ideals ... subset= I_2 subset= I_1 subset= I_0=R, such that I_iI_j subset= I_(i+j) for all indices i,j. ...

A division algebra, also called a "division ring" or "skew field," is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not ...

A proper subgroup is a proper subset H of group elements of a group G that satisfies the four group requirements. "H is a proper subgroup of G" is written H subset G. The ...

A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. ...

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism. Moreover, if G and G^' are isomorphic, ...

Let a group G have a group presentation G=<x_1,...,x_n|r_j(x_1,...,x_n),j in J> so that G=F/R, where F is the free group with basis {x_1,...,x_n} and R is the normal subgroup ...

Frey's theorem states that every Abelian category is a subcategory of some category of modules over a ring. Mitchell (1964) has strengthened this, saying every Abelian ...

A short exact sequence of groups A, B, and C is given by two maps alpha:A->B and beta:B->C and is written 0->A->B->C->0. (1) Because it is an exact sequence, alpha is ...