On uniformly $1$-absorbing primary ideals

In this article, we introduce the concept of uniformly 1-absorbing primary ideal which is a generalization of uniformly primary ideal. Let $R$ be a commutative ring with a unity and $P$ be a proper ideal of $R$. $P$ is said to be a uniformly 1-absorbing primary ideal if there exists $N\in \mathbb{N}$ and whenever $xyz\in P$ for some nonunits $x,y,z\in R$, then either $xy\in P$ or $z^{N}\in P$. The smallest aforementioned $N\in\mathbb{N}$ is called the order of $P$ and denoted by $\operatorname{ord}_{R}(P)=N$. In addition to giving many properties of uniformly 1-absorbing primary ideals, we investigate the relationship between uniformly 1-absorbing primary ideals and other classical ideals such as uniformly primary ideals and 1-absorbing primary ideals.