For a commutative ring R, we investigate the structural and ideal-theoretic properties of the quotient polynomial ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R[x]/\langle x^n \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for each positive integer n, where R[x] denotes the ring of polynomials in an indeterminate x, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\langle x^n \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> is the ideal generated by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we determine the units and the Jacobson radical of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R[x]/\langle x^n \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, providing foundational insight into its local structure. Furthermore, we identify necessary and sufficient conditions under which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R[x]/\langle x^n \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">⟨</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mo stretchy="false">⟩</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfies various ideal-theoretic properties, including quasi-primary ideals, (quasi) J-ideals, r-ideals, (quasi) n-ideals, 2-absorbing ideals, and 2-absorbing primary ideals. Our results extend and unify numerous existing findings, and illustrative examples are provided to clarify and delimit the theoretical framework.

Quotient polynomial rings and some of their ideals

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发表日期：

2026年5月25日

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Kamal Paykan

摘要

For a commutative ring R, we investigate the structural and ideal-theoretic properties of the quotient polynomial ring $R[x]/\langle x^n \rangle $ $R [x] / ⟨ x^{n} ⟩$ for each positive integer n, where R[x] denotes the ring of polynomials in an indeterminate x, and $\langle x^n \rangle $ $⟨ x^{n} ⟩$ is the ideal generated by $x^n$ $x^{n}$ . In this paper, we determine the units and the Jacobson radical of $R[x]/\langle x^n \rangle $ $R [x] / ⟨ x^{n} ⟩$ , providing foundational insight into its local structure. Furthermore, we identify necessary and sufficient conditions under which $R[x]/\langle x^n \rangle $ $R [x] / ⟨ x^{n} ⟩$ satisfies various ideal-theoretic properties, including quasi-primary ideals, (quasi) J-ideals, r-ideals, (quasi) n-ideals, 2-absorbing ideals, and 2-absorbing primary ideals. Our results extend and unify numerous existing findings, and illustrative examples are provided to clarify and delimit the theoretical framework.