论文标题
美元
$\mathcal{P}_1$-covers over commutative rings
论文作者
论文摘要
在本文中，我们考虑$ \ MATHCAL {P} _1（r）投影尺寸模块的$，最多是一个通勤环$ r $，我们调查$ \ Mathcal {p} _1（r）$是封面。更确切地说，我们研究了Enochs对此类别的猜想，这是$ \ Mathcal {p} _1（r）$覆盖的问题一定意味着$ \ Mathcal {p} _1（r）$在直接限制下关闭。在交换性的半卫星环$ r $的情况下，我们肯定地回答了这个问题。这给出了一个cotorsion对$（\ Mathcal {p} _1（r），\ Mathcal {p} _1（r）^\ perp）$的示例，这不一定是有限类型的，因此$ \ nathcal {p} _1（p} _1（r）$满足Enochs的猜想。此外，我们描述了$ \ varinjlim \ natercal {p} _1（r）$ over（不必要的交换）环，这些戒指允许经典的商圈。
In this paper we consider the class $\mathcal{P}_1(R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when $\mathcal{P}_1(R)$ is a covering class. More precisely, we investigate Enochs' Conjecture for this class, that is the question of whether $\mathcal{P}_1(R)$ is covering necessarily implies that $\mathcal{P}_1(R)$ is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring $R$. This gives an example of a cotorsion pair $(\mathcal{P}_1(R), \mathcal{P}_1(R)^\perp)$ which is not necessarily of finite type such that $\mathcal{P}_1(R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $\varinjlim \mathcal{P}_1(R)$ over (not-necessarily commutative) rings which admit a classical ring of quotients.
