Some Results on $n$-Stable Rings

Amir M. Rahimi
2003 Missouri Journal of Mathematical Sciences  
All rings are commutative rings with identity, and J(R) denotes the Jacobson radical of the ring R. For any fixed integer n ≥ 1, it is shown that the class of all n-stable rings is properly contained in the class of all n + 1-stable rings. Results are given showing the connection between several types of rings whose finite sequences satisfy different stability conditions, some involving J(R). It is shown that in the strongly n-stable case, it suffices to check whether the n + 1tuples satisfy
more » ... stable condition. In addition to other results and an example of a ring which is not n-stable for any integer n ≥ 1, examples are given to show the distinction between the different types of stability cases. Finally, in the last section, some surjective mapping properties of a generalized form of GL n (R) and SL n (R) in connection to some stable conditions in the ring R are investigated.
doi:10.35834/2003/1502129 fatcat:c36vfe2s5zfpbmzrfwpuu7grxy