Note that a two-sided ideal I is regular if the

quotient algebra A/I has [a] = a + I as a unit element.

This leads the description of the projective class ring [r.sub.p]([H.sub.n](q)), the Jacobson radical J([R.sub.p]([H.sub.n](q))) of the projective class algebra [R.sub.p] ([H.sub.n](q)) and the

quotient algebra [R.sub.p] ([H.sub.n] (q))/ J ([R.sub.p] ([H.sub.n] (q))).

Then there exists a homogeneous [H.sub.n](0)-invariant ideal [I.sub.[alpha]] of the multigraded algebra F[[B.sub.n]] such that the

quotient algebra F[[B.sub.n]]/[I.sub.[alpha]] becomes a projective [H.sub.n](0)-module with multigraded noncommutative characteristic equal to

Let L([H.sub.1](X)) be the Lie algebra over Z defined to be the

quotient algebra of T([H.sub.1](X)) by the ideal I generated by elements of the form

Filters are closely related with lattice valued logics, they are also particularly interesting because they closely related to congruence relation, with any filter we can associate a

quotient algebra. If we want such a

quotient algebra to satisfy particular properties, we need to put extra conditions on the filter.

In this case, the

quotient algebra A/k(A) is a nonzero unital Abelian [C.sup.*]-algebra.

equivalently, those ideals J [subset] k[x] for which G spans the

quotient algebra k[x]/J.

A topological algebra A over C is a Gelfand-Mazur algebra if the

quotient algebra A/M is topologically isomorphic to C for every M [member of] m(A).

Let [??] = Fin(M)/I be the

quotient algebra. We let [??] = x + I.

However, there are several complications to be overcome: (i) the definition of the join matrix, (ii) the choice of the finiteness condition on the join matrices, and (iii) the choice of the definition of the

quotient algebra.

Then all non-zero elements in the

quotient algebra A/M are topologically invertible.

For example, the algebra QSym is the

quotient algebra of FQSym where [F.sub.[sigma]] and [F.sub.[tau]] are identified if [sigma] and [tau] have the same descents.