Communications of the
Korean Mathematical Society CKMS

Let $R$ be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi $J$-ideals lying properly between the class of $J$-ideals and the class of quasi $J$-ideals. A proper ideal $I$ of $R$ is called a strongly quasi $J$-ideal if, whenever $a$, $b\in R$ and $ab\in I$, then $a^{2}\in I$ or $b\in {\rm Jac}(R)$. Firstly, we investigate some basic properties of strongly quasi $J$-ideals. Hence, we give the necessary and sufficient conditions for a ring $R$ to contain a strongly quasi $J$-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi $J$-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.

Keywords: $J$-ideals, quasi $J$-ideals, strongly quasi $J$-ideals

MSC numbers: Primary 13A15, 13A99