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In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by ...
Any two-sided ideal of a ring R is an R-R-bimodule, with the ring multiplication both as the left and as the right multiplication. Any module over a commutative ring R has the natural structure of a bimodule. For example, if M is a left module, we can define multiplication on the right to be the same as multiplication on the left.
Cutting along a 1-sided manifold may make a non-orientable manifold orientable – such as cutting along an equator of the real projective plane – but may not, such as cutting along a 1-sided curve in a higher genus non-orientable surface, maybe the simplest example of this is seen when one cut a mobius band along its core curve.
The one-sided shift S acting on is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator S is a compression of T −1 , in the sense that T − 1 y = S x for each x ∈ ℓ 2 ( N ) , {\displaystyle T^{-1}y=Sx{\text{ for each }}x\in \ell ^{2}(\mathbb {N} ),} where y is the vector in ℓ ...
In algebra, a division ring, also called a skew field (or, occasionally, a sfield [1] [2]), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring [3] in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a –1, such that a a –1 = a –1 a = 1.
In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If a −1 is the left inverse of a, then a ∗ b = a ∗ c implies a −1 ∗ (a ∗ b) = a −1 ∗ (a ∗ c), which implies b = c by associativity. For example, every quasigroup, and thus every group, is cancellative.
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For any , is a two-sided -module, and the direct sum decomposition is a direct sum of -modules. R {\displaystyle R} is an associative R 0 {\displaystyle R_{0}} -algebra . An ideal I ⊆ R {\displaystyle I\subseteq R} is homogeneous , if for every a ∈ I {\displaystyle a\in I} , the homogeneous components of a ...