Relationship of certain rings of infinite matrices over integers

Mario Petrich; Pedro V. Silva

Displaying similar documents to “Relationship of certain rings of infinite matrices over integers”

Left APP-property of formal power series rings

Zhongkui Liu, Xiao Yan Yang (2008)

Archivum Mathematicum

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A ring $R$ is called a left APP-ring if the left annihilator $l_{R} (R a)$ is right $s$ -unital as an ideal of $R$ for any element $a \in R$ . We consider left APP-property of the skew formal power series ring $R [[x; α]]$ where $α$ is a ring automorphism of $R$ . It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R [[x; α]]$ is left APP if and only if for any sequence $(b_{0}, b_{1}, \dots)$ of elements of $R$ the ideal $l_{R}$ $(\sum_{j = 0}^{\infty} \sum_{k = 0}^{\infty} R α^{k} (b_{j}))$ is right $s$ -unital. As an application we give a sufficient condition under which...

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

Vijay Kumar Bhat (2013)

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 $σ$ be an automorphism of $R$ and $δ$ a $σ$ -derivation of $R$ . Then $R$ is said to be an almost $δ$ -divided ring if every minimal prime ideal of $R$ is $δ$ -divided. Let $R$ be a Noetherian ring which is also an algebra over $ℚ$ ( $ℚ$ is the field of rational numbers). Let $σ$ be an automorphism of $R$ such that $R$ is a $σ (*)$ -ring and $δ$ a $σ$ -derivation of $R$ such that $σ (δ (a)) = δ (σ (a))$ for all $a \in R$ . Further,...

On a theorem of McCoy

Rajendra K. Sharma, Amit B. Singh (2024)

Mathematica Bohemica

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We study McCoy’s theorem to the skew Hurwitz series ring $(HR, ω)$ for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring $R$ satisfies McCoy’s theorem of skew Hurwitz series.

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let $m > 1, s \geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p (x) \geq 0, q = q (x) \geq 0, n = n (x) \geq 0, r = r (x) \geq 0$ such that either $x^{p} [x^{n}, y] x^{q} = x^{r} [x, y^{m}] y^{s}$ or $x^{p} [x^{n}, y] x^{q} = y^{s} [x, y^{m}] x^{r}$ for all $y \in R$ . In the present paper it is shown that $R$ is commutative if it satisfies the property $Q (m)$ (i.e. for all $x, y \in R, m [x, y] = 0$ implies $[x, y] = 0$ ).

A canonical directly infinite ring

Mario Petrich, Pedro V. Silva (2001)

Czechoslovak Mathematical Journal

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Let $ℕ$ be the set of nonnegative integers and $ℤ$ the ring of integers. Let $ℬ$ be the ring of $N \times N$ matrices over $ℤ$ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $ℬ$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $ℱ$ consisting...