t-Invertibility and Comparability
Robert Gilmer, Joe Leonard Mott, Muhammad Zafrullah · 2023
Let D be an integral domain with quotient field K and let F ( D ) denote the set of nonzero fractional ideals of D . For https://www.w3.org/1998/Math/MathML" display="inline"> A ∈ F ( D ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2593.tif "/> we define https://www.w3.org/1998/Math/MathML" display="inline"> A − 1 = { x ∈ K ∣ x A ⊆ D } https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2594.tif "/> . An ideal https://www.w3.org/1998/Math/MathML" display="inline"> A ∈ F ( D ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2595.tif "/> is t -invertible if (i) https://www.w3.org/1998/Math/MathML" display="inline"> ( A A − 1 ) − 1 = D https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2596.tif "/> , (ii) there exists https://www.w3.org/1998/Math/MathML" display="inline"> x 1 , x 2 , … , x n ∈ A https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2597.tif "/> such that https://www.w3.org/1998/Math/MathML" display="inline"> ( x 1 , … , x n ) − 1 = A − 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2598.tif "/> , and (iii) https://www.w3.org/1998/Math/MathML" display="inline"> A − 1 = ( ( y 1 , … , y m ) − 1 ) − 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2599.tif "/> for some https://www.w3.org/1998/Math/MathML" display="inline"> y 1 , … , y m ∈ K https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003421917/80230c9a-c5b4-433b-a203-0f6b6ddf13a0/content/eq2600.tif "/> . The notion of t -invertibility has been useful in characterizing Krull domains in various ways [ HZ ], [ J ], [ K ] and [ MZ ].