## Displaying similar documents to “Real holomorphy rings and the complete real spectrum”

### Conditions under which $R\left(x\right)$ and $R〈x〉$ are almost Q-rings

Archivum Mathematicum

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All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$-ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$-ring is a ring whose localization at every prime ideal is a $Q$-ring. In this paper, we first prove that the statements, $R$ is an almost $ZPI$-ring and $R\left[x\right]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R\left(x\right)$ is an extension of a prime ideal of $R$, the ring $R$...

### Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Czechoslovak Mathematical Journal

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Let $R$ be a ring. We recall that $R$ is called a near pseudo-valuation ring if every minimal prime ideal of $R$ is strongly prime. Let now $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. Then $R$ is said to be an almost $\delta$-divided ring if every minimal prime ideal of $R$ is $\delta$-divided. Let $R$ be a Noetherian ring which is also an algebra over $ℚ$ ($ℚ$ is the field of rational numbers). Let $\sigma$ be an automorphism of $R$ such that $R$ is a $\sigma \left(*\right)$-ring and $\delta$ a $\sigma$-derivation of $R$ such that $\sigma \left(\delta \left(a\right)\right)=\delta \left(\sigma \left(a\right)\right)$ for all $a\in R$. Further,...

### Derivations with Engel conditions in prime and semiprime rings

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d\left[x,y\right]\right)}^{m}={\left[x,y\right]}_{n}$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${\left[d\left(x\right),d\left(y\right)\right]}_{m}={\left[x,y\right]}^{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

### Isolated points and redundancy

Commentationes Mathematicae Universitatis Carolinae

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We describe the isolated points of an arbitrary topological space $\left(X,\tau \right)$. If the $\tau$-specialization pre-order on $X$ has enough maximal elements, then a point $x\in X$ is an isolated point in $\left(X,\tau \right)$ if and only if $x$ is both an isolated point in the subspaces of $\tau$-kerneled points of $X$ and in the $\tau$-closure of $\left\{x\right\}$ (a special case of this result is proved in Mehrvarz A.A., Samei K., , J. Sci. Islam. Repub. Iran (1999), no. 3, 193–196). This result is applied to an arbitrary subspace of the prime spectrum $Spec\left(R\right)$ of...

### $SV$-Rings and $SV$-Porings

Annales de la faculté des sciences de Toulouse Mathématiques

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$SV$-rings are commutative rings whose factor rings modulo prime ideals are valuation rings. $SV$-rings occur most naturally in connection with partially ordered rings (= porings) and have been studied only in this context so far. The present note first develops the theory of $SV$-rings systematically, without assuming the presence of a partial order. Particular attention is paid to the question of axiomatizability (in the sense of model theory). Partially ordered $SV$-rings ($SV$-porings)...