Journal of Algebra and Its Applications, Feb 23, 2023
A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a stro... more A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of [Formula: see text], called the class of quasi-[Formula: see text]-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of [Formula: see text] is a maximal or a minimal ideal of [Formula: see text]. Many examples are given to illustrate the obtained results.
Journal of Algebra and Its Applications, May 27, 2019
Let [Formula: see text] be commutative ring with [Formula: see text]. In this paper, we introduce... more Let [Formula: see text] be commutative ring with [Formula: see text]. In this paper, we introduce a subclass of the class of [Formula: see text]-absorbing primary ideals called the class of [Formula: see text]-ideals. An ideal [Formula: see text] of [Formula: see text] with [Formula: see text] is called [Formula: see text]-ideal if whenever [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. Many examples and results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals, the 2-absorbing primary ideals, and the [Formula: see text]-ideals. Also, we use the [Formula: see text]-ideals to characterize some kind of rings.
In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we cal... more In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we call 2 r 2r -ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2 r 2r -ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2 r 2r -ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2 r 2r -ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A A -property.
A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a stro... more A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of [Formula: see text], called the class of quasi-[Formula: see text]-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of [Formula: see text] is a maximal or a minimal ideal of [Formula: see text]. Many examples are given to illustrate the obtained results.
Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-abso... more Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyz ∈ I for some nonunits x, y, z ∈ R, then xy ∈ I or z ∈ I. It is well known that prime ideal ⇒ 1-absorbing prime ideal ⇒ primary ideal ⇒ semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings.
Journal of Algebra and Its Applications, Feb 23, 2023
A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a stro... more A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of [Formula: see text], called the class of quasi-[Formula: see text]-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of [Formula: see text] is a maximal or a minimal ideal of [Formula: see text]. Many examples are given to illustrate the obtained results.
Journal of Algebra and Its Applications, May 27, 2019
Let [Formula: see text] be commutative ring with [Formula: see text]. In this paper, we introduce... more Let [Formula: see text] be commutative ring with [Formula: see text]. In this paper, we introduce a subclass of the class of [Formula: see text]-absorbing primary ideals called the class of [Formula: see text]-ideals. An ideal [Formula: see text] of [Formula: see text] with [Formula: see text] is called [Formula: see text]-ideal if whenever [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. Many examples and results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals, the 2-absorbing primary ideals, and the [Formula: see text]-ideals. Also, we use the [Formula: see text]-ideals to characterize some kind of rings.
In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we cal... more In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we call 2 r 2r -ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2 r 2r -ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2 r 2r -ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2 r 2r -ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A A -property.
A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a stro... more A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then [Formula: see text] or [Formula: see text] (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of [Formula: see text], called the class of quasi-[Formula: see text]-ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of [Formula: see text] is a maximal or a minimal ideal of [Formula: see text]. Many examples are given to illustrate the obtained results.
Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-abso... more Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyz ∈ I for some nonunits x, y, z ∈ R, then xy ∈ I or z ∈ I. It is well known that prime ideal ⇒ 1-absorbing prime ideal ⇒ primary ideal ⇒ semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings.
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