Editing Glossary of ring theory (section)

== A ==
{{glossary}}

{{term|1=Amitsur complex}}
{{defn|1=The [[Amitsur complex]] of a ring homomorphism is a cochain complex that measures the extent in which the ring homomorphism fails to be [[faithfully flat ring homomorphism|faithfully flat]].}}

{{term|1=Artinian}}
{{defn|1=A left [[Artinian ring]] is a ring satisfying the [[descending chain condition]] for left ideals; a right Artinian ring is one satisfying the descending chain condition for right ideals. If a ring is both left and right Artinian, it is called ''Artinian''. Artinian rings are Noetherian rings.}}

{{term|1=associate}}
{{defn|1=In a commutative ring, an element ''a'' is called an [[Divisibility (ring theory)|associate]] of an element ''b'' if ''a'' divides ''b'' and ''b'' divides ''a''.}}

{{term|1=automorphism}}
{{defn|1=A [[ring automorphism]] is a ring isomorphism between the same ring; in other words, it is a unit element of the endomorphism ring of the ring that is multiplicative and preserves the multiplicative identity.}}
{{defn|1=An [[algebra automorphism]] over a commutative ring ''R'' is an algebra isomorphism between the same algebra; it is a ring automorphism that is also ''R''-linear.}}

{{term|1=Azumaya}}
{{defn|1=An [[Azumaya algebra]] is a generalization of a central simple algebra to a non-field base ring.}}

{{glossary end}}