Difference between revisions of "Nil ideal"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/N066/N.0606670 Nil ideal | ||
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| − | + | A subset $ A $ | |
| + | of a ring $ R $ | ||
| + | is called nil if each element of it is nilpotent (cf. [[Nilpotent element|Nilpotent element]]). An ideal of $ R $ | ||
| + | is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that | ||
| + | |||
| + | $$ | ||
| + | \mathop{\rm Jac} ( R) \supset \textrm{ Nil Rad } ( R) \supset \textrm{ Prime Rad } ( R), | ||
| + | $$ | ||
| + | |||
| + | where $ \mathop{\rm Jac} ( R) $ | ||
| + | denotes the [[Jacobson radical|Jacobson radical]] of $ R $ | ||
| + | and $ \textrm{ Prime Rad } ( R) $ | ||
| + | is the prime radical of $ R $, | ||
| + | i.e. the intersection of all prime ideals of $ R $. | ||
| + | Each of the inclusions can be proper. If $ R $ | ||
| + | is commutative, $ \textrm{ Nil Rad } ( R) = \textrm{ Prime Rad } ( R) $. | ||
| + | The prime radical is also called the lower nil radical, and the nil radical the upper nil radical. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Faith, "Algebra" , '''II. Ring theory''' , Springer (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.H. Rowen, "Ring theory" , '''1''' , Acad. Press (1988)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Faith, "Algebra" , '''II. Ring theory''' , Springer (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.H. Rowen, "Ring theory" , '''1''' , Acad. Press (1988)</TD></TR></table> | ||
Latest revision as of 08:02, 6 June 2020
A subset $ A $
of a ring $ R $
is called nil if each element of it is nilpotent (cf. Nilpotent element). An ideal of $ R $
is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that
$$ \mathop{\rm Jac} ( R) \supset \textrm{ Nil Rad } ( R) \supset \textrm{ Prime Rad } ( R), $$
where $ \mathop{\rm Jac} ( R) $ denotes the Jacobson radical of $ R $ and $ \textrm{ Prime Rad } ( R) $ is the prime radical of $ R $, i.e. the intersection of all prime ideals of $ R $. Each of the inclusions can be proper. If $ R $ is commutative, $ \textrm{ Nil Rad } ( R) = \textrm{ Prime Rad } ( R) $. The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.
References
| [a1] | C. Faith, "Algebra" , II. Ring theory , Springer (1976) |
| [a2] | J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) |
| [a3] | L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) |
How to Cite This Entry:
Nil ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_ideal&oldid=12426
Nil ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_ideal&oldid=12426