In the mid-1970s, famous American homological algebra and commutative algebra mathematician Maurice Auslander (August 3, 1926 - November 18, 1994) introduced the Auslander algebra. Ever since, the concept has been widely used in the algebra branch of mathematics. Basically, the Auslander algebra of an algebra A is the endomorphism ring of the total of the indecomposable modules of A.
If the projective dimension of its socle does not exceed one
If the projective dimension of its socle is more than one
If the projective dimension of its socle is between 1 and -1
If the projective dimension of its socle equals to one t
1973
1974
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1976
Field
Variable
Natural number
Kepler value
Variable
Trigonometric value
Time
Valency
Real number
Whole number
Natural number
Constant number
Exceeds 1
Less than -1
Does not exceed 1
More than -1
If the projective dimension of its socle does not exceed one
If the projective dimension of its socle is more than one
If the projective dimension of its socle is between 1 and -1
If the projective dimension of its socle equals to one t
Aqua-hereditary
Quasi hereditary
Hereditary
Non-hereditary
Γ ≤ 2
Γ = 2
Γ < 2
Γ > 2
Endomorphism ring
Exomorphism ring
Static ring
External ring
Decomposable modules
Incompressible modules
Normal modules
Indecomposable modules