Search results
Results From The WOW.Com Content Network
Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module. If M n ( R ) is the ring of n × n matrices over a ring R , M is an M n ( R )-module, and e i is the n × n matrix with 1 in the ( i , i ) -entry (and zeros elsewhere), then e i M is an ...
Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.
The added structure must be compatible, in some sense, with the algebraic structure. Topological group: a group with a topology compatible with the group operation. Lie group: a topological group with a compatible smooth manifold structure. Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order.
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible.Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.
For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
The smallest friendly number is 6, forming for example, the friendly pair 6 and 28 with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved ...
Equinumerosity is compatible with the basic set operations in a way that allows the definition of cardinal arithmetic. [1] Specifically, equinumerosity is compatible with disjoint unions: Given four sets A, B, C and D with A and C on the one hand and B and D on the other hand pairwise disjoint and with A ~ B and C ~ D then A ∪ C ~ B ∪ D.
Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called bialgebras, and in fact most of the important coalgebras considered in practice are bialgebras. Examples of coalgebras include the tensor algebra, the exterior algebra, Hopf algebras and Lie bialgebras. Unlike the ...