An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and ... more An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and 0 ? ab ? I, then a ? I or b ? ?I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ? R and 0 ? abc ? I, then ab ? I or c ? ?I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary.
Let R be a commutative ring and let n > 1 be an integer. We introduce a simple graph, denoted by ... more Let R be a commutative ring and let n > 1 be an integer. We introduce a simple graph, denoted by Γt(Mn(R)), which we call the trace graph of the matrix ring Mn(R), such that its vertex set is Mn(R) * and such that two distinct vertices A and B are joined by an edge if and only if Tr(AB) = 0 where Tr(AB) denotes the trace of the matrix AB. We prove that Γt(Mn(R)) is connected with diam(Γt(Mn(R))) = 2 and gr(Γt(Mn(R))) = 3. We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γt(Mn(R)). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given ...
Bulletin of the Iranian Mathematical Society, 2018
Let $$f:A\rightarrow B$$f:A→B be a homomorphism of commutative rings and let J be an ideal of B. ... more Let $$f:A\rightarrow B$$f:A→B be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of $$A\times B$$A×B given by $$A\bowtie ^fJ=\{(a,f(a)+j)\mid a\in A, \, j\in J\}$$A⋈fJ={(a,f(a)+j)∣a∈A,j∈J}. In this paper, we give some characterizations for the amalgamation construction to be a quasi-Frobenius ring.
Let R be commutative ring with . A proper ideal I of R is called a 1-absorbing primary ideal of R... more Let R be commutative ring with . A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements and , then or . It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize rings R over which every semi-primary ideal is 1-absorbing primary and (resp. Noetherian) rings R over which every 1-absorbing primary ideal is prime (resp. primary). Many examples are given to illustrate the obtained results.
Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1... more Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ 0. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN -rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.
In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commu... more In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.
We define a particular case of a FPI-ring called a weak FPI-ring. We investigate the transfer of ... more We define a particular case of a FPI-ring called a weak FPI-ring. We investigate the transfer of the weak FPI-ring to trivial ring extensions, pullbacks, subring retracts, amalgamated duplication of a ring along an ideal, and direct product of rings.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2021
Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, w... more Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.
Pythagorean fuzzy set (PFS) introduced by Yager (2013) is the extension of intuitionistic fuzzy s... more Pythagorean fuzzy set (PFS) introduced by Yager (2013) is the extension of intuitionistic fuzzy set (IFS) introduced by Atanassov (1983). PFS is also known as IFS of type-2. Pythagorean fuzzy soft set (PFSS), introduced by Peng et al. (2015) and later studied by Guleria and Bajaj (2019) and Naeem et al. (2019), are very helpful in representing vague information that occurs in real world circumstances. In this article, we introduce the notion of Pythagorean fuzzy soft topology (PFS-topology) defined on Pythagorean fuzzy soft set (PFSS). We define PFS-basis, PFS-subspace, PFS-interior, PFS-closure and boundary of PFSS. We introduce Pythagorean fuzzy soft separation axioms, Pythagorean fuzzy soft regular and normal spaces. Furthermore, we present an application of PFSSs to multiple criteria group decision making (MCGDM) using choice value method in the real world problems which yields the optimum results for investment in the stock exchange. We also render an application of PFS-topolog...
In this paper, we define the new idea of triangular cubic hesitant fuzzy number (TCHFN). We discu... more In this paper, we define the new idea of triangular cubic hesitant fuzzy number (TCHFN). We discuss some basic operational laws of triangular cubic hesitant fuzzy number and hamming distance of TCHFNs. We introduce the new concept of triangular cubic hesitant TOPSIS method. Furthermore, we extend the classical cubic hesitant the technique for order of preference by similarity to ideal solution (TOPSIS) method to solve the Multi-Criteria decision-making (MCDM) method based on triangular cubic hesitant TOPSIS method. The new ranking method for TCHFNs is used to rank the alternatives. Finally, an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.
Abstract In the first part of this paper, we point out some errors in Communications in Algebra, ... more Abstract In the first part of this paper, we point out some errors in Communications in Algebra, 45(11) (2017), 4631–4645 and then we give a correct version of the errant results. The second part presents our contribution in the study of two-sided α-derivations satisfying certain identities over near-rings.
Indian Journal of Pure and Applied Mathematics, 2020
In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an... more In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R , and d be a derivation of R . If [[ x , d ( x )], y ] ∈ P for all x , y ∈ R , then d ( R ) ⊆ P or R / P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R , we prove that either R is commutative or there exists a minimal prime ideal P of R such that d ( R ) ⊆ P . As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d ( P ) ⊆ P .
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and ... more An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and 0 ? ab ? I, then a ? I or b ? ?I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ? R and 0 ? abc ? I, then ab ? I or c ? ?I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary.
Let R be a commutative ring and let n > 1 be an integer. We introduce a simple graph, denoted by ... more Let R be a commutative ring and let n > 1 be an integer. We introduce a simple graph, denoted by Γt(Mn(R)), which we call the trace graph of the matrix ring Mn(R), such that its vertex set is Mn(R) * and such that two distinct vertices A and B are joined by an edge if and only if Tr(AB) = 0 where Tr(AB) denotes the trace of the matrix AB. We prove that Γt(Mn(R)) is connected with diam(Γt(Mn(R))) = 2 and gr(Γt(Mn(R))) = 3. We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γt(Mn(R)). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given ...
Bulletin of the Iranian Mathematical Society, 2018
Let $$f:A\rightarrow B$$f:A→B be a homomorphism of commutative rings and let J be an ideal of B. ... more Let $$f:A\rightarrow B$$f:A→B be a homomorphism of commutative rings and let J be an ideal of B. The amalgamation of A with B along J with respect to f is the subring of $$A\times B$$A×B given by $$A\bowtie ^fJ=\{(a,f(a)+j)\mid a\in A, \, j\in J\}$$A⋈fJ={(a,f(a)+j)∣a∈A,j∈J}. In this paper, we give some characterizations for the amalgamation construction to be a quasi-Frobenius ring.
Let R be commutative ring with . A proper ideal I of R is called a 1-absorbing primary ideal of R... more Let R be commutative ring with . A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements and , then or . It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize rings R over which every semi-primary ideal is 1-absorbing primary and (resp. Noetherian) rings R over which every 1-absorbing primary ideal is prime (resp. primary). Many examples are given to illustrate the obtained results.
Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1... more Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ 0. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN -rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.
In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commu... more In this note, we indicate some errors in [S. Ali, N. A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 2016, 1, 9–14] and present the correct versions of the erroneous results.
We define a particular case of a FPI-ring called a weak FPI-ring. We investigate the transfer of ... more We define a particular case of a FPI-ring called a weak FPI-ring. We investigate the transfer of the weak FPI-ring to trivial ring extensions, pullbacks, subring retracts, amalgamated duplication of a ring along an ideal, and direct product of rings.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2021
Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, w... more Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.
Pythagorean fuzzy set (PFS) introduced by Yager (2013) is the extension of intuitionistic fuzzy s... more Pythagorean fuzzy set (PFS) introduced by Yager (2013) is the extension of intuitionistic fuzzy set (IFS) introduced by Atanassov (1983). PFS is also known as IFS of type-2. Pythagorean fuzzy soft set (PFSS), introduced by Peng et al. (2015) and later studied by Guleria and Bajaj (2019) and Naeem et al. (2019), are very helpful in representing vague information that occurs in real world circumstances. In this article, we introduce the notion of Pythagorean fuzzy soft topology (PFS-topology) defined on Pythagorean fuzzy soft set (PFSS). We define PFS-basis, PFS-subspace, PFS-interior, PFS-closure and boundary of PFSS. We introduce Pythagorean fuzzy soft separation axioms, Pythagorean fuzzy soft regular and normal spaces. Furthermore, we present an application of PFSSs to multiple criteria group decision making (MCGDM) using choice value method in the real world problems which yields the optimum results for investment in the stock exchange. We also render an application of PFS-topolog...
In this paper, we define the new idea of triangular cubic hesitant fuzzy number (TCHFN). We discu... more In this paper, we define the new idea of triangular cubic hesitant fuzzy number (TCHFN). We discuss some basic operational laws of triangular cubic hesitant fuzzy number and hamming distance of TCHFNs. We introduce the new concept of triangular cubic hesitant TOPSIS method. Furthermore, we extend the classical cubic hesitant the technique for order of preference by similarity to ideal solution (TOPSIS) method to solve the Multi-Criteria decision-making (MCDM) method based on triangular cubic hesitant TOPSIS method. The new ranking method for TCHFNs is used to rank the alternatives. Finally, an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.
Abstract In the first part of this paper, we point out some errors in Communications in Algebra, ... more Abstract In the first part of this paper, we point out some errors in Communications in Algebra, 45(11) (2017), 4631–4645 and then we give a correct version of the errant results. The second part presents our contribution in the study of two-sided α-derivations satisfying certain identities over near-rings.
Indian Journal of Pure and Applied Mathematics, 2020
In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an... more In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R , and d be a derivation of R . If [[ x , d ( x )], y ] ∈ P for all x , y ∈ R , then d ( R ) ⊆ P or R / P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R , we prove that either R is commutative or there exists a minimal prime ideal P of R such that d ( R ) ⊆ P . As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d ( P ) ⊆ P .
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the ... more In this paper, we attempt to construct a class of Armendariz-Like properties. We investigate the transfer of the Armendariz-Like properties to trivial ring extensions to localization and direct product of rings, and then generate new families of rings with zero-divisors subject to some given Armendariz-like properties. The article includes a brief discussion of the scope and precision of our results.
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