### Generalization of primal superideals

#### Abstract

Let \(R\) be a commutative super-ring with unity \(1\not=0\). A proper superideal of \(R\) is a superideal \(I\) of \(R\) such that \(I\not=R\). Let \(\phi : \mathfrak{I}(R)\rightarrow\mathfrak{I}(R)\cup\{\emptyset\}\) be any function, where \(\mathfrak{I}(R)\) denotes the set of all proper superideals of \(R\). A homogeneous element \(a\in R\) is \(\phi\)-{\it prime} to \(I\) if \(ra\in I-\phi(I)\) where \(r\) is a homogeneous element in \(R\), then \(r\in I\). We denote by \(\nu_\phi(I)\) the set of all homogeneous elements in \(R\) that are not \(\phi\)-prime to \(I\). We define \(I\) to be \(\phi\)-\textit{primal} if the set \[P=\begin{cases}[(\nu_\phi(I))_0+(\nu_\phi(I))_1\cup\{0\}]+\phi(I) & :\quad {\rm if}\ \phi\not=\phi_\emptyset\\ (\nu_\phi(I))_0+(\nu_\phi(I))_1& :\quad {\rm if}\ \phi=\phi_\emptyset\end{cases}\]

forms a superideal of \(R\). For example if we take \(\phi_\emptyset(I)=\emptyset\) (resp. \(\phi_0(I)=0\)), a \(\phi\)-\textit{primal} superideal is a primal superideal (resp., a weakly primal superideal). In this paper we study several generalizations of primal superideals of \(R\) and their properties.

#### Keywords

#### Full Text:

PDF#### References

Ameer Jaber, Central simple superalgebras with anti-automorphisms of order two of the ﬁrst kind, J. Algebra, 323, 7 (2010) 18491859.

Ameer Jaber, Central simple superalgebras with superantiautomorphism of order two of the second kind, Turkish Journal of Mathematics, 35 (2011), 11-21.

Ameer Jaber, Division Z3-Algebras, International Electronic Journal of Algebra, 7 (2010), 1-11.

Ameer Jaber, Existence of Pseudo-Superinvolutions of the First Kind, International Journal of Mathematics and Mathematical Sciences, Article ID 386468, 12 pages doi:10.1155/2008/386468.

Ameer Jaber, Primitive Z3-algebras with Z3-involution, Far East Journal of Mathematical Sciences, 48 (2011), no. 2, 225-244.

Ameer Jaber, Product of graded submodules, Turkish Journal of Mathematics, 35 (2011) , 1 12.

Ameer Jaber, ∆-supergraded Submodules, International Mathematical Forum, 5 (2010), 22, 1091-1104.

Ameer Jaber, Weakly Primal Graded Superideals, Tamkang Journal of Mathematics, 43 (2012), 1, 123-135.

D. Anderson, M. Bataineh, Generalizations of prime ideals, Comm. in Algebra 36 (2008), 686-696.

Ebrahimi Atani, Youseﬁan Darani, On Weakly Primal Ideals(I), Demonstratio Math. 40 (2007), 23-32.

L. Fuchs, On Primal ideals, Amer. Math. Soc. 1 (1950), 1-6.

Y. A. Bahturin, A. Giambruno, Group Gradings on associative algebras with involution, DOI:10.4153/CMB-2008-020-7, Canad. Math. Bull., 51 (2008), 182-194.

Youseﬁan Darani, Generalizations of primal ideals in commutative rings, MATEMATIQKI VESNIK, 64 (2012), 1, 2531.

### Refbacks

- There are currently no refbacks.