%0 Journal Article
%T The Morita Equivalence for the <math xmlns='http://www.w3.org/1998/Math/MathML'> <mrow>  <msub>   <mi>A</mi>   <mi>&#x221E;</mi>  </msub>  </mrow></math>-Algebras
%A Hawazin Daif Allah Alzahrani
%J Advances in Pure Mathematics
%P 483-490
%@ 2160-0384
%D 2025
%I Scientific Research Publishing
%R 10.4236/apm.2025.157023
%X This paper explores the foundational and advanced aspects of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>A</mi>
   <mi>&#x221E;</mi>
  </msub>
  </mrow>
</math>

-algebras. We present a comprehensive study of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>A</mi>
   <mi>&#x221E;</mi>
  </msub>
  </mrow>
</math>

-algebras and their homology, emphasizing the construction of long exact sequences in simplicial homology and their implications for short exact sequences of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>A</mi>
   <mi>&#x221E;</mi>
  </msub>
  </mrow>
</math>

-algebras. Furthermore, we investigate the trace and inclusion maps in matrix <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>A</mi>
   <mi>&#x221E;</mi>
  </msub>
  </mrow>
</math>

-algebras, proving their mutual invertibility and highlighting their role in preserving homological properties. These results underscore the utility of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>A</mi>
   <mi>&#x221E;</mi>
  </msub>
  </mrow>
</math>

-algebras in simplifying complex algebraic systems while maintaining essential structural invariants.
%K A-Infinity
%K Homology
%K Inclusion Map
%K Trace Map
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=144175