<p>We explore the concept of flatness in <i>S</i>-acts, inspired by module theory. While in some cases flat acts align with the exactness of the tensor functor, we establish that, in general, the flatness of acts is weaker than the exactness of the tensor functor. This leads to the notion of exact flatness. We characterize the class of exact flat acts as those that are both flat and indecomposable. Furthermore, we examine conditions under which, in a Rees short exact sequence <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A_S {\mathop {\longrightarrow }\limits ^{f}} B_S {\mathop {\longrightarrow }\limits ^{g}} C_S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>S</mi> </msub> <mover> <mo stretchy="false">⟶</mo> <mi>f</mi> </mover> <msub> <mi>B</mi> <mi>S</mi> </msub> <mover> <mo stretchy="false">⟶</mo> <mi>g</mi> </mover> <msub> <mi>C</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, the exact flatness of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C_S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> implies that of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B_S$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation>, and conversely. We also present some classifications of monoids based on the exact flatness properties of their acts.</p>

Exact flat acts over monoids

期刊论文
发表日期：

2025年9月3日

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Hamid Rasouli,
Reza Aminizadeh

摘要

We explore the concept of flatness in S-acts, inspired by module theory. While in some cases flat acts align with the exactness of the tensor functor, we establish that, in general, the flatness of acts is weaker than the exactness of the tensor functor. This leads to the notion of exact flatness. We characterize the class of exact flat acts as those that are both flat and indecomposable. Furthermore, we examine conditions under which, in a Rees short exact sequence $A_S {\mathop {\longrightarrow }\limits ^{f}} B_S {\mathop {\longrightarrow }\limits ^{g}} C_S$ $A_{S} \overset{f}{⟶} B_{S} \overset{g}{⟶} C_{S}$ , the exact flatness of $A_S$ $A_{S}$ and $C_S$ $C_{S}$ implies that of $B_S$ $B_{S}$ , and conversely. We also present some classifications of monoids based on the exact flatness properties of their acts.