The Jacobson Density Theorem and Applications

1.1. Strictly Cyclic Modules and Modular Right Ideals. For a ring A with identity, cyclic modules are precisely those of the form a\A where a is a right ideal. What might be a useful analogous statement for a ring without identity? This question motivates what follows in this subsection. A module M is strictly cyclic if there exists m in M such that mA = M (such an m is called a generator); a right ideal a is modular if there exists e in A such that a− ea belongs to a for every a in A (such an e is a left identity for a). These two notions are related thus: strictly cyclic modules are precisely those of the form a\A for modular right ideals a. Proof: the image of a left identity for a in a\A is a generator; if m is a generator for M , then any element e such that me = m is a left identity for the right ideal consisting of annihilators of m. The usefulness of the notion of a modular right ideal is further borne out by the following basic observations. To set these up, first note that the annihilator of a module is a two sided ideal. Thus, for modules N ⊆ M , the “colon” (N : M) := {a ∈ A |Ma ⊆ N} is a two sided ideal in A. In the case when M is A and N a right ideal a, we can compare a with (a : A). For a modular right ideal a with left identity e, • (a : A) ⊆ a; in fact, (a : A) is the largest two sided ideal contained in a. • There is no proper right ideal containing both a and e. Thus, by an application of Zorn’s Lemma, a is contained in a maximal (proper) right ideal (which is necessarily modular—see the next item). • e continues to be a left identity for right ideals containing a.