… regular Hausdorff space, and C(X) will denote the ring of all real-… isomorphisms between
proper ideals of rings of continuous functions. The main lemma is that for any such isomorphism …

… isomorphism between a lattice of ideals (called the ideals with local unit) of an arbitrary ring
R and the lattice of all ideals of a ring R… if R is bi-regular in the sense of Arens and Kaplansky …

… Hence, according to Theorem 1, we see that there exists a regular element ao such that
ao=a (mod r(L)) ie xao =xa (=x~) for any element x in L. Thus we have our assertion. …

… It is also shown that a commutative regular ring is n-strongly invariant for all positive integers
n… [ence, T is an isomorphism with inverse T'. The nil radical of a commutative ring 0 is the set …

… 6.13] constructed two von Neumann regular rings A and B … to results on isomorphisms of
endomorphism rings which are … that any isomorphism of (strongly dense sub) rings of faithful …

… Hence S<p* = Sd>, and 5 is regular. It remains to be shown that 6 is a ring isomorphism.
Clearly ty = 0 implies "F = 0, and 6 is onto by Lemma 3. If ^x and tyt are in 5 then (\IVFi)- = MM^i …

… quasi-regular. We insert 0 in the spectrum of x unless A has a unit element and x is regular.
… Before stating the next lemma, we consider the following situation: <f> is a ring isomorphism …

… of regular rings which are not one-sided unit regular. Finally (Theorem 5), we prove that a
one-sided unit regular ring … (In fact, any isomorphism of Px onto P2 can be extended to such an …

… We devote the remainder of this section to calculating the Jacobson radical of the incidence
ring I(X, A). Since we will use the quasi-regular characterization of the radical, wewill first …

… We determine the isomorphisms, anti-isomorphisms and derivations for such rings. In Chapter
V … This is clear since the regular representation of a division ring is faithful and irreducible. …