We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Radicals of semirings.
- Authors
Sharma, Ram Parkash; Sharma, Richa; Madhu
- Abstract
It is shown that the classes γ Σ 0 = { R | Σ 0 R = ϕ } and γ Σ 1 = { R | Σ 1 R = ϕ } of semirings are radical classes, where Σ 0 R is the class of subtractive-simple right R -semimodules and Σ 1 R is the class of right R -semimodules isomorphic to R / ρ I for some maximal-subtractive right ideal I of R. We define the lower Jacobson Bourne radical rad l (R) ⊆ ker (Σ 0 R) and upper Jacobson Bourne radical rad u (R) ⊆ ker (Σ 1 R) of R. For a semiring R , rad l (R) ⊆ rad (R) holds, where rad (R) is the Jacobson Bourne radical of R. The radical rad l (R) = ker (Σ 0 R) and also coincides with rad (R) , if we restrict the class Σ 0 R to additively cancellative semimodules. The upper radical rad u (R) = ker (Σ 1 R) and rad (R) ⊆ rad u (R) , if R is additively cancellative. Further, rad u (R / ρ rad u (R)) = { 0 } , if R is a commutative semiring with 1 R. The subtractive-primitiveness and subtractive-semiprimitiveness of R are closely related to the upper radical rad u (R). Finally, we show that rad u -semisimplicity of semirings are Morita invariant property with some restrictions.
- Subjects
SEMIRINGS (Mathematics); JACOBSON radical; RADICAL theory
- Publication
Asian-European Journal of Mathematics, 2020, Vol 13, Issue 07, pN.PAG
- ISSN
1793-5571
- Publication type
Article
- DOI
10.1142/S1793557120501387