In this paper, we prove that s-perspective, a-perspective and ad-perspective conditions are equivalent conditions, for finite dimensional left R-modules, where R is an K-algebra. We construct more examples of s-perspective modules and hence a-perspective modules which do not have essentially unique direct complements. We introduce a stronger variation of s-perspective modules named as ss-perspective. Let R be any ring and let M be any left R-module. We will say that M is ss-perspective if for any isomorphic two direct summands A and B of M, the equalities <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M=A\oplus X = B\oplus Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <mi>A</mi> <mo>⊕</mo> <mi>X</mi> <mo>=</mo> <mi>B</mi> <mo>⊕</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> imply that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A=B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X=Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove that the class of ss-perspective modules is contained properly in the class of s-perspective modules. We also prove that for commutative rings, s-perspectivity and ss-perspectivity coincide. Finally, we prove that for a left R-module M with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S=\textrm{End}_R(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <msub> <mtext>End</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> is ss-perspective then M is ss-perspective, while we prove that if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$_SS$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>S</mi> <mrow /> </mmultiscripts> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> is ss-perspective and M is weak duo, then M is ss-perspective.

On the Classes of Almost (Dual) Perspective Modules

期刊论文
发表日期：

2026年6月19日

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Gabriella D’Este,
Derya Keskin Tütüncü

摘要

In this paper, we prove that s-perspective, a-perspective and ad-perspective conditions are equivalent conditions, for finite dimensional left R-modules, where R is an K-algebra. We construct more examples of s-perspective modules and hence a-perspective modules which do not have essentially unique direct complements. We introduce a stronger variation of s-perspective modules named as ss-perspective. Let R be any ring and let M be any left R-module. We will say that M is ss-perspective if for any isomorphic two direct summands A and B of M, the equalities $M=A\oplus X = B\oplus Y$ $M = A \oplus X = B \oplus Y$ imply that $A=B$ $A = B$ or $X=Y$ $X = Y$ . We prove that the class of ss-perspective modules is contained properly in the class of s-perspective modules. We also prove that for commutative rings, s-perspectivity and ss-perspectivity coincide. Finally, we prove that for a left R-module M with $S=\textrm{End}_R(M)$ $S = {End}_{R} (M)$ , if $S_S$ $S_{S}$ is ss-perspective then M is ss-perspective, while we prove that if $_SS$ $_{S} S$ is ss-perspective and M is weak duo, then M is ss-perspective.