| Summary: | A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let <i>R</i> be a finite chain ring with invaraints <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><msup><mi>k</mi><mo>′</mo></msup><mo>,</mo><mi>m</mi><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> the automorphism group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> of <i>R</i> is known. The main purpose of this article is to study the structure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> First, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of <i>R</i> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∤</mo><mi>k</mi><mo>.</mo></mrow></semantics></math></inline-formula> In addition, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is investigated under a more general condition; that is, <i>R</i> is very pure and <i>p</i> need not divide <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>.</mo></mrow></semantics></math></inline-formula> Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.
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