a non-zerodivisor in the center of R, then xR is a dense right ideal of R. If I is a two-sided ideal of R, I is dense as a right ideal if and only if the left annihilator of I is zero, that is, . In particular in commutative rings, the dense ideals are precisely the ideals which are faithful modules. Applications Rational hull of a module Every right R module M has a maximal essential extension E(M) which is its injective hull. The analogous construction using a maximal dense extension results in the rational hull Ẽ(M) which is a submodule of E(M). When a module has no proper rational extension, so that Ẽ(M) = M, the module is said to be rationally complete. If R is right nonsingular, then of course Ẽ(M) = E(M). The rational hull is readily identified within the injective hull. Let S=EndR(E(M)) be the endomorphism ring of the injective hull. Then an element x of the injective hull is in the rational hull if and only if x is sent to zero by all maps in S which are zero on M. In symbols, In general, there may be maps in S which are zero on M and yet are nonzero for some x not in M, and such an x would not be in the rational hull. Maximal right ring of quotients The maximal right ring of quotients can be described in two ways in connection with dense right ideals of R. In one method, Ẽ(R) is shown to be module isomorphic to a certain endomorphism ring, and the ring structure is taken across this isomorphism to imbue Ẽ(R) with a ring structure, that of the maximal right ring of quotients. In a second method, the maximal right ring of quotients is identified with a set of equivalence classes of homomorphisms from dense right ideals of R into R. The equivalence relation says that two functions are equivalent if they agree on a dense right ideal of R. References Module theory Ring theory