The Jacobson radical " m " of a local ring " R " ( which is equal to the unique maximal left ideal and also to the unique maximal right ideal ) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of " R ".
42.
Every basis gives rise to cell chains ( one for each topological ordering of \ Lambda ) and choosing a basis of every left ideal \ Delta / J _ { k-1 } \ subseteq J _ k / J _ { k-1 } one can construct a corresponding cell basis for A.
43.
If " D " is a division ring and \ sigma is a ring endomorphism which is not an automorphism, then the skew polynomial ring D [ x, \ sigma ] is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring.
44.
Rings which are simple as rings but not as field does not have any nontrivial ideals ( since any ideal of M ( n, " R " ) is of the form M ( n, " I " ) with I an ideal of R ), but has nontrivial left ideals ( namely, the sets of matrices which have some fixed zero columns ).
