Left Ideal

Some articles on left ideal, left, left ideals, ideal, ideals:

Divisible Group - Generalization
... zero-divisor, and some authors require that R is a domain.) For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M ... module is also called principally injective module.) For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism ... Since injective left modules extend homomorphisms from all left ideals to R, injective modules are clearly divisible in sense 2 and 3 ...
Von Neumann Regular Ring - Facts
... The following statements are equivalent for the ring R R is von Neumann regular every principal left ideal is generated by an idempotent every finitely generated left ideal is generated by an idempotent every ... every short exact sequence of left R-modules is pure exact The corresponding statements for right modules are also equivalent to R being von Neumann regular ... dimension 0 and is reduced Every localization of R at a maximal ideal is a field R is a subring of a product of fields closed under taking "weak inverses" of x∈R (the unique element y ...
Algebra Over A Field - Basic Concepts - Subalgebras and Ideals
... A left ideal of a K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the ... In symbols, we say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements ... closed under scalar multiplication), 3) z · x is in L (L is closed under left multiplication by arbitrary elements) ...
Matrix Ring - Structure
... There is a one-to-one correspondence between the two-sided ideals of Mn(R) and the two-sided ideals of R ... Namely, for each ideal I of R, the set of all n×n matrices with entries in I is an ideal of Mn(R), and each ideal of Mn(R) arises in this way ... For n ≥ 2, not every left ideal or right ideal of Mn(R) arises by the previous construction from a left ideal or a right ideal in R ...
Essential Extension
... submodule H of M, implies that As a special case, an essential left ideal of R is a left ideal which is essential as a submodule of the left module RR ... The left ideal has non-zero intersection with any non-zero left ideal of R ... Analogously, and essential right ideal is exactly an essential submodule of the right R module RR ...

Famous quotes containing the words ideal and/or left:

    It is well worth the efforts of a lifetime to have attained knowledge which justifies an attack on the root of all evil—viz. the deadly atheism which asserts that because forms of evil have always existed in society, therefore they must always exist; and that the attainment of a high ideal is a hopeless chimera.
    Elizabeth Blackwell (1821–1910)

    Cannon to right of them,
    Cannon to left of them,
    Cannon in front of them
    Volleyed and thundered;
    Alfred Tennyson (1809–1892)