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In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also examine the preservation of diameter of zero-divisor graph under content extensions.

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions...

In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I+J)(I\cap J)=IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family...