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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.
[3] The usage primarily comes from translations of Euclid's Elements, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not ...
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 ...
Hasse diagram of the natural numbers, partially ordered by "x≤y if x divides y".The numbers 4 and 6 are incomparable, since neither divides the other. In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true.
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.
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting complex matrices A, B with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) can be matched up as in such a way that the multiset of eigenvalues of any polynomial (,) in the two matrices is the multiset of the ...
[3] The set of submodules of a given module M, together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a lattice that satisfies the modular law: Given submodules U, N 1, N 2 of M such that N 1 ⊆ N 2, then the following two submodules are equal: (N 1 + U) ∩ N 2 = N 1 + (U ∩ N 2).
3 + 2. Note that the ancient Sanskrit sages discovered many years before Fibonacci that the number of compositions of any natural number n as the sum of 1's and 2's is the nth Fibonacci number! Note that these are not general compositions as defined above because the numbers are restricted to 1's and 2's only. 1=1 (1) 2=1+1=2 (2) 3=1+1+1=1+2=2 ...