In this article, We prove the following results: (1). We show that if R is an affine C-algebra of dimension 4, where C is either a subfield F of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\overline{\mathbb {F}}_p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi mathvariant="double-struck">F</mi> <mo>¯</mo> </mover> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C=\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>=</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>. Then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\text {WMS}_3(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>WMS</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an abelian group. (2). Let R be a commutative ring with sdim(<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R)\le 3,~n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>3</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that if [v]/[w] is the quotient of orbits <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$[v],~[w]\in \frac{\text {Um}_n(R)}{\text {E}_n(R)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">[</mo> <mi>v</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">]</mo> </mrow> <mo>∈</mo> <mfrac> <mrow> <msub> <mtext>Um</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <msub> <mtext>E</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, then [w]/[v] is the inverse of [v]/[w].

A group structure on $\text {WMS}_3(R)$

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发表日期：

2026年4月13日

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Gopal Sharma,
Sampat Sharma

摘要

In this article, We prove the following results: (1). We show that if R is an affine C-algebra of dimension 4, where C is either a subfield F of ${\overline{\mathbb {F}}_p}$ ${\bar{F}}_{p}$ or $C=\mathbb {Z}$ $C = Z$ . Then $\text {WMS}_3(R)$ ${WMS}_{3} (R)$ is an abelian group. (2). Let R be a commutative ring with sdim( $R)\le 3,~n\ge 3$ $R) \leq 3, n \geq 3$ . We show that if [v]/[w] is the quotient of orbits $[v],~[w]\in \frac{\text {Um}_n(R)}{\text {E}_n(R)}$ $[v], [w] \in \frac{{Um}_{n} (R)}{E_{n} (R)}$ , then [w]/[v] is the inverse of [v]/[w].

A group structure on \(\text {WMS}_3(R)\)

摘要