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* An idempotent in the quotient ring R / I is said to lift modulo I if there is an idempotent f in R such that.
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idempotent and ring
An idempotent element of a ring is, by definition, an element that is idempotent for the ring's multiplication.
* A ring is semisimple if and only if every right ( or every left ) ideal is generated by an idempotent.
* A ring is von Neumann regular if and only if every finitely generated right ( or every finitely generated left ) ideal is generated by an idempotent.
* A ring for which the annihilator every subset S of R is generated by an idempotent is called a Baer ring.
* A ring R can be written as with each e < sub > i </ sub > a local idempotent if and only if R is a semiperfect ring.
If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.
If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.
If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.
Thus for a ring in which 2 is invertible, the idempotent elements correspond to involutions in a one-to-one manner.
In fact, the projections of a vector space are exactly the idempotent elements of the ring of linear transformations of the vector space.
idempotent and R
So in particular, every central idempotent a in R gives rise to a decomposition of R as a direct sum of the corner rings aRa and.
idempotent and /
However, placing an order for a car for the customer is typically not idempotent, since running the method / call several times will lead to several orders being placed.
idempotent and I
idempotent and is
Methods GET, HEAD, OPTIONS and TRACE, being prescribed as safe, should also be idempotent, as HTTP is a stateless protocol.
In contrast, the POST method is not necessarily idempotent, and therefore sending an identical POST request multiple times may further affect state or cause further side effects ( such as financial transactions ).
* A unary operation ( or function ) is idempotent if, whenever it is applied twice to any value, it gives the same result as if it were applied once ; i. e.,.
* A binary operation is idempotent if, whenever it is applied to two equal values, it gives that value as the result.
* Given a binary operation, an idempotent element ( or simply an idempotent ) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result.
A unary operation f, that is, a map from some set S into itself, is called idempotent if, for all x in S,
In particular, the identity function id < sub > S </ sub >, defined by, is idempotent, as is the constant function K < sub > c </ sub >, where c is an element of S, defined by.
A unary operation is idempotent if it maps each element of S to a fixed point of f. For a set with n elements there are
Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if
In particular an identity element of ★, if it exists, is idempotent with respect to the operation ★.
idempotent and said
e is the two-sided ideal e R. For each indecomposable R-module, there only one such primitive idempotent that does not annihilate it, and the module is said to belong to ( or to be in ) the corresponding block ( in which case, all its composition factors also belong to that block ).
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