Let $R$ be a commutative ring with $1\ne 0$. A proper ideal $I$ of $R$ is said to be a strongly quasi-primary ideal if, whenever $a,b\in R$ with $ab\in I$, then either $a^2\in I$ or $b\in \sqrt{I}$. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of $R$ is prime. Many examples are given to illustrate the obtained results.

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关于强准初级理想的注释

让$R$是一个交换环$1\ne 0$. 适当的理想$我$的$R$被认为是一个强烈的准主理想，如果$\mathrm{一个},b\in R$和$\mathrm{一个}b\in 我$，那么要么${\mathrm{一个}}^2\in 我$或者$b\in \sqrt{我}$. 在本文中，我们描述了诺特环和约化环，在这些环上，每个（分别为非零）真理想都是强准主的。我们还刻画了每一个强烈的准初级理想的环$R$是素数。给出了许多例子来说明得到的结果。