Difference between revisions of "Quasi-regular ring"
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| − | A [[ | + | A [[ring]] in which every element is quasi-regular. An element $a$ of an alternative (in particular, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that |
$$a+a'+aa'=a+a'+a'a=0.$$ | $$a+a'+aa'=a+a'+a'a=0.$$ | ||
| − | The element $a'$ is called the quasi-inverse of $a$. If $R$ is a ring with identity 1, then an element $a\in R$ is quasi-regular with quasi-inverse $a'$ if and only if the element $1+a$ is invertible in $R$ with inverse $1+a'$. Every [[Nilpotent element|nilpotent element]] is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [[#References|[2]]]. | + | The element $a'$ is called the quasi-inverse of $a$. If $R$ is a [[ring with identity]] 1, then an element $a\in R$ is quasi-regular with quasi-inverse $a'$ if and only if the element $1+a$ is invertible in $R$ with inverse $1+a'$. Every [[Nilpotent element|nilpotent element]] is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the ring of (non-commutative) [[formal power series]] without constant terms. There exist simple associative quasi-regular rings [[#References|[2]]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Sasiada, P.M. Cohn, "An example of a simple radical ring" ''J. of Algebra'' , '''5''' : 3 (1967) pp. 373–377</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> E. Sasiada, P.M. Cohn, "An example of a simple radical ring" ''J. of Algebra'' , '''5''' : 3 (1967) pp. 373–377</TD></TR> | ||
| + | </table> | ||
Latest revision as of 21:27, 6 September 2017
A ring in which every element is quasi-regular. An element $a$ of an alternative (in particular, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that
$$a+a'+aa'=a+a'+a'a=0.$$
The element $a'$ is called the quasi-inverse of $a$. If $R$ is a ring with identity 1, then an element $a\in R$ is quasi-regular with quasi-inverse $a'$ if and only if the element $1+a$ is invertible in $R$ with inverse $1+a'$. Every nilpotent element is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [2].
References
| [1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
| [2] | E. Sasiada, P.M. Cohn, "An example of a simple radical ring" J. of Algebra , 5 : 3 (1967) pp. 373–377 |
Quasi-regular ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-regular_ring&oldid=31927