Let R be a commutative ring with identity and M be an R-module. For a non-negative integer d and an ideal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frak{b}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> </math></EquationSource> </InlineEquation> of R, the concept of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(d,\frak{b})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-injectivity in the category of R-modules is defined. We characterize the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(d,\frak{b})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-injective hull of a module M as a submodule of E(M). Then we focus on the case when the ring R is Noetherian and see the connection with some modules caused by some ideal transform functor <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T_{(d,\frak{b})}(-)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and some local cohomology functors <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H_{(d,\frak{b})}^{i}(-)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> based on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(d,\frak{b})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>. As results we will see that over a Noetherian ring the functors <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma_{(d,\frak{b})}(-)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$T_{(d,\frak{b})}(-)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> preserve the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(d,\frak{b})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-injectivity. Among other results, for a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(d,\frak{b})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-torsion free module M, we find a condition under which <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$T_{(d,\frak{b})}(M)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>T</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mrow> <mi mathvariant="fraktur">b</mi> </mrow> <mo mathvariant="fraktur" stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is injective.

$(d,\frak{b})$-injectivity and some ideal transforms

期刊论文
发表日期：

2026年2月10日

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Ahad Ghaffari,
Naser Zamani,
Mirsadegh Sayedsadeghi

摘要

Let R be a commutative ring with identity and M be an R-module. For a non-negative integer d and an ideal $\frak{b}$ $b$ of R, the concept of $(d,\frak{b})$ $(d, b)$ -injectivity in the category of R-modules is defined. We characterize the $(d,\frak{b})$ $(d, b)$ -injective hull of a module M as a submodule of E(M). Then we focus on the case when the ring R is Noetherian and see the connection with some modules caused by some ideal transform functor $T_{(d,\frak{b})}(-)$ $T_{(d, b)} (-)$ and some local cohomology functors $H_{(d,\frak{b})}^{i}(-)$ $H_{(d, b)}^{i} (-)$ based on $(d,\frak{b})$ $(d, b)$ . As results we will see that over a Noetherian ring the functors $\Gamma_{(d,\frak{b})}(-)$ $Γ_{(d, b)} (-)$ and $T_{(d,\frak{b})}(-)$ $T_{(d, b)} (-)$ preserve the $(d,\frak{b})$ $(d, b)$ -injectivity. Among other results, for a $(d,\frak{b})$ $(d, b)$ -torsion free module M, we find a condition under which $T_{(d,\frak{b})}(M)$ $T_{(d, b)} (M)$ is injective.

\((d,\frak{b})\)-injectivity and some ideal transforms

摘要