Since injective left modules extend homomorphisms from " all " left ideals to " R ", injective modules are clearly divisible in sense 2 and 3.
22.
The above definition is satisfied if " R " has a finite number of maximal right ideals ( and finite number of maximal left ideals ).
23.
The left ideal " I " can be viewed as a right-module over, and the ring is clearly isomorphic to the algebra of homomorphisms on this module.
24.
An " ideal " ( sometimes called a " two-sided ideal " for emphasis ) is a subgroup which is both a left ideal and a right ideal.
25.
A two-sided ideal is a left ideal that is also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals.
26.
In other words, if " I " is a minimal left ideal, then, where " e " is the idempotent matrix with 1 in the entry and zero elsewhere.
27.
An internal characterization of left primitive rings is as follows : a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals.
28.
R is called left semiartinian if _ { R } R is semiartinian, that is, R is left semiartinian if for any left ideal I, R / I contains a simple submodule.
29.
However, this latter order is not a maximal one, and therefore ( as it turns out ) less suitable for developing a theory of left ideals comparable to that of algebraic number theory.
30.
If " x " is in " R ", then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by " x ".
