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Two elements x and y of a set S are said to be commutative under a binary operation * if they satisfy x*y=y*x. (1) Real numbers are commutative under addition x+y=y+x (2) and ...

Let A denote an R-algebra, so that A is a vector space over R and A×A->A (1) (x,y)|->x·y. (2) Now define Z={x in A:x·y=0 for some y in A!=0}, (3) where 0 in Z. An Associative ...

A ring is commutative if the multiplication operation is commutative.

A monoid that is commutative i.e., a monoid M such that for every two elements a and b in M, ab=ba. This means that commutative monoids are commutative, associative, and have ...

A commutative diagram is a collection of maps A_i-->^(phi_i)B_i in which all map compositions starting from the same set A and ending with the same set B give the same ...

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal. If K is a field, the ...

The application of characteristic p methods in commutative algebra, which is a synthesis of some areas of commutative algebra and algebraic geometry.

Two algebraic objects that are commutative, i.e., A and B such that A*B=B*A for some operation *, are said to commute with each other.

The notion of height is defined for proper ideals in a commutative Noetherian unit ring R. The height of a proper prime ideal P of R is the maximum of the lengths n of the ...

A regular ring in the sense of commutative algebra is a commutative unit ring such that all its localizations at prime ideals are regular local rings. In contrast, a von ...