Narrow Results

It is well known that a commutative ring R is complemented (that is, given a ∈ R there exists b ∈ R such that ab = 0 and a + b is a regular element) if and only if the total ring of quotients of R is von Neumann regular. We consider generalizations of the notion of a complemented ring and their implications for the total ring of quotients. We then look at the specific case when the ring is a ring of continuous real-valued functions on a topological space.

In this paper, we discuss several constructions that lead to new examples of nil-clean, clean and exchange rings. Extensions by ideals contained in the Jacobson radical is the common theme of these constructions. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations, we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean and exchange rings, arising from poset algebras are also discussed.

In this paper, we continue to study the differential inverse power series ring $R[[x^{-1};\delta ]]$, where $R$ is a ring equipped with a derivation $\delta$. We characterize when $R[[x^{-1};\delta ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$-ring, respectively. Furthermore, we prove that $R[[x^{-1};\delta ]]$ is a domain satisfying the $ACC$ on principal left ideals if and only if so does $R$. Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R[[x^{-1};\delta ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[x^{-1};\delta ]]$. Our results extend and unify many existing results.

In the present note, we continue the study of skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ and skew inverse power series ring $R[[x^{-1};\alpha ,\delta ]]$, where $R$ is a ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Necessary and sufficient conditions are obtained for $R[[x^{-1};\alpha ,\delta ]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and $I$-ring, respectively. It is shown here that $R((x^{-1};\alpha ,\delta ))$ (respectively $R[[x^{-1};\alpha ,\delta ]]$) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does $R$. Also, we investigate the problem when a skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ has the same Goldie rank as the ring $R$ and is proved that, if $R$ is a semiprime right Goldie ring, then $R((x^{-1};\alpha ,\delta ))$ is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring $R$ and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when $f(x)\in R((x^{-1};\alpha ,\delta ))$ is nilpotent.

Using the equivalence of unique generation and cleanness of $C(X)$, we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.  66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math.  34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if $C(X)$ is $\mathrm{\text{UG}}$ (uniquely generated) then $M_n(C(X))$ is too, and $R$ is strongly associate if and only if $R[[x]]$ is, where $R=C(X)$. We give topological characterizations of $\mathrm{\text{UG}}$ and $\mathrm{\text{AS}}$ (annihilator-stable) elements of $C(X)$ for continuum spaces $X$ and using this, we observe that the product of two UG elements need not be $\mathrm{\text{UG}}$. It is shown that the set of elements in $C(X)$ which have stable range 1 and the set of $\mathrm{\text{AS}}$ elements of $C(X)$ coincide and several examples are given which show that the set of $\mathrm{\text{UG}}$ elements, the set of $\mathrm{\text{AS}}$ elements and the set of clean elements of $C(X)$ can differ. Finally, we characterize spaces $X$ for which every clean element of $C(X)$ or every element of $C(X)$ which has stable range 1 is $\mathrm{\text{UG}}$.