The Maximal Regular Ideal of Some Commutative Rings
The Maximal Regular Ideal of Some Commutative Rings
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发表时间：
2006
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通讯作者：
E. A. Osba;M. Henriksen;Osama Alkam;F. Smith
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作者：
E. A. Osba;M. Henriksen;Osama Alkam;F. Smith
1950年，在《美国数学学会学报》第一卷中，B。Brown和N.麦考伊表明，每一个（不一定是交换）环$R$有一个理想$\frak M（R）$组成的元素$a$，其中有一个$x$，使$axa=a$，并最大关于这一性质。仅考虑$R$是交换的且有单位元的情况，通常不容易确定$\frak M（R）$何时不只是零理想。我们确定在许多情况下何时发生这种情况：即当$a$或$1-a$中至少有一个具有冯诺依曼逆时，当$R$是局部环的乘积时（例如，当$R$是$\Bbb Z_{n}$或$\Bbb Z_{n}[i]$时，当$R$是多项式环或幂级数环时，当$R$是拓扑空间上所有实值连续函数的环时。
In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $\frak M (R)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $\frak M (R)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse, when $R$ is a product of local rings (e.g., when $R$ is $\Bbb Z_{n}$ or $\Bbb Z_{n}[i]$), when $R$ is a polynomial or a power series ring, and when $R$ is the ring of all real-valued continuous functions on a topological space.