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A semiring is a set together with two binary operators S(+,*) satisfying the following conditions: 1. Additive associativity: For all a,b,c in S, (a+b)+c=a+(b+c), 2. Additive ...

The term "similarity transformation" is used either to refer to a geometric similarity, or to a matrix transformation that results in a similarity. A similarity ...

An abstract vector space of dimension n over a field k is the set of all formal expressions a_1v_1+a_2v_2+...+a_nv_n, (1) where {v_1,v_2,...,v_n} is a given set of n objects ...

There are at least two definitions of hypercomplex numbers. Clifford algebraists call their higher dimensional numbers hypercomplex, even though they do not share all the ...

A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p ...

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral ...

Abstract Algebra

A group whose group operation is identified with multiplication. As with normal multiplication, the multiplication operation on group elements is either denoted by a raised ...

Two matrices A and B which satisfy AB=BA (1) under matrix multiplication are said to be commuting. In general, matrix multiplication is not commutative. Furthermore, in ...

A fractional ideal is a generalization of an ideal in a ring R. Instead, a fractional ideal is contained in the number field F, but has the property that there is an element ...