Let n be an integer co-prime to 3 and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_662_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$ {\mathbb {Z}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be the ring of integers modulo n. In this article, we study the structure and generators of the unit group of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_662_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$ {\mathbb {Z}}_nC_3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> <msub> <mi>C</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Further, if <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_662_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$T_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> denotes the elementary abelian 3-group of order <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_662_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$3^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>3</mn> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation>, then we provide the structure of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_662_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$ {\mathbb {U}}({\mathbb {Z}}_nT_m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">U</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> <msub> <mi>T</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also solve the normal complement problem in each case.

On the units in group rings over ${\mathbb {Z}}_n$

期刊论文
发表日期：

2025年7月22日

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Himanshu Setia,
Surinder Kaur,
Manju Khan

摘要

Let n be an integer co-prime to 3 and let $ {\mathbb {Z}}_n$ $Z_{n}$ be the ring of integers modulo n. In this article, we study the structure and generators of the unit group of $ {\mathbb {Z}}_nC_3 $ $Z_{n} C_{3}$ . Further, if $T_m$ $T_{m}$ denotes the elementary abelian 3-group of order $3^m$ $3^{m}$ , then we provide the structure of $ {\mathbb {U}}({\mathbb {Z}}_nT_m)$ $U (Z_{n} T_{m})$ . We also solve the normal complement problem in each case.

On the units in group rings over \({\mathbb {Z}}_n\)

摘要