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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.
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 ...
A two-sided property is fulfilled on both sides. A one-sided property is related to one (unspecified) of two sides. Although the terms are similar, left–right distinction in algebraic parlance is not related either to left and right limits in calculus , or to left and right in geometry .
If e is an idempotent element (e 2 = e) of an associative algebra A, the two-sided Peirce decomposition of A given the single idempotent e is the direct sum of eAe, eA(1 − e), (1 − e)Ae, and (1 − e)A(1 − e). There are also corresponding left and right Peirce decompositions.
A compound of two "line segment" digons, as the two possible alternations of a square (note the vertex arrangement). The apeirogonal hosohedron , containing infinitely narrow digons. Any straight-sided digon is regular even though it is degenerate, because its two edges are the same length and its two angles are equal (both being zero degrees).
In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity. To see this, note that if l is a left identity and r is a right identity, then l = l ∗ r = r.
Bears receiver Keenan Allen said that issues ran deeper than that and went back to the offseason. “Too nice of a guy," Allen said, according to Kalyn Kahler of ESPN, via Dan Wiederer of the ...
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...