Prime Ideals in General Rings

Prime ideals in nonassociative rings

“…We shall prove that .4" is the intersection of all M-prime ideals which contain A. This, of course, generalizes the corresponding theorem for the associative case which was established in [7]. The special case for prime ideals (u = xix2) in a general ring has also been proved by Amitsur [l] and by Behrens [4].…”

“…Inasmuch as the proofs follow easily by the methods of [7], we state the following two theorems without proof.…”

On Properties of Uniformly Strongly Fuzzy Ideals

“…Moreover, I is USPf ideal if its complement is a t-system. Definition 15 [18] A subset M of a ring R is called an m-system if for any two elements x, y∈ M there exists r ∈ R such that xry∈ M. [3] A subset T of a ring R is called a t-system if there exists a finite set F  R such that for any two elements x, y∈T there exists f ∈ F such that xfy∈ T.…”

“…So, it will be called the insulator of T. Note that, the empty set will be a t-system. Proposition 17 [18] If M is a t-system, then M is a m-system. Proposition 18 [18] I is a prime ideal of a ring R if R \ I (the complement of I in R) is an m-system.…”

Sur les anneaux premiers noethériens à gauche

“…On sait que cette propriété est équivalente au fait que le produit de deux idéaux bilatères non nuls soit un idéal bilatère non nul ou encore au fait que le produit de deux idéaux à gauche non nuls soit un idéal à gauche non nul (cf. N. H. McCoy [11]). La notion d'anneau d'intégrité, c'est-à-dire d'anneau sans diviseurs de zéro (anneau tel que la condition ab = o implique a = o ou b = o) est un cas particulier de celle d'anneau premier.…”