On a class of lattice-ordered rings
On a class of lattice-ordered rings
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关于一类格序环
DOI：
10.1090/s0002-9939-1957-0089179-6
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发表时间：
1957
期刊：
影响因子：
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通讯作者：
B. Brainerd
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文献类型：
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作者：
B. Brainerd
对于某个真实的数X，符号V表示格的最小上界。任何环R是正则环[10]，如果对每个xE：R存在x 0 ER使得xx 0x =x。显然，每个正则F-环R包含一个极大有界子F-环X，所有xER的F-环满足方程（1.1）。正则F-环和它的极大有界子F-环之间的关系类似于完全正则空间X上的所有连续函数的环和X上的所有有界连续函数的环之间的关系。例如，定理3证明了R的极大理想与W的极大理想之间存在一一对应。(For连续函数环的理论，见[5]和[6]。环R的极大理想M是真实的[6]，如果商环RM与真实的域环同构。F-环R的理想S是闭的，如果战神，n>1且Vn 1anER蕴涵Vl. janES.定理5和定理6证明了正则F-环的闭极大理想是真实的，且正则F-环R的闭极大理想与R的极大有界子F-环W的闭极大理想一一对应。它是中野[9，pp. 39，212]证明了有界F-环是紧Hausdorff空间上的环，并且格同构于所有连续函数的环。因此，每个有界F-环都是半单真实的Banach代数.这里使用的“真实的”是经典意义上的，即偏序环R是
for some real number X, the symbol V denoting the lattice least upper bound. Any ring R is regular [10] if for each xE:R there is an x0ER such that xx0x=x. It is evident that every regular F-ring R contains a maximal bounded sub-F-ring X, the F-ring of all xER satisfying equation (1.1). The relationship between a regular F-ring and its maximal bounded sub-F-ring is analogous to that between the ring of all continuous functions on a completely regular space X and the ring of all bounded continuous functions on X. For example, it is shown in Theorem 3 that there is a one-to-one correspondence between the maximal ideals of R and those of W. (For the theory of rings of continuous functions, see [5] and [6].) A maximal ideal M of a ring R is real [6] if the quotient ring R M is ring-isomorphic to the real field. An ideal S of an F-ring R is closed if aRES, n>1 and Vn1anER imply Vl.janES. It is proved in Theorems 5 and 6 that the closed maximal ideals of a regular F-ring are real and that there is a one-to-one correspondence between the closed maximal ideals of a regular F-ring R and the closed maximal ideals of W, the maximal bounded sub-F-ring of R. It is a direct corollary of some results of Nakano [9, pp. 39, 212] that a bounded F-ring is ringand lattice-isomorphic to the ring of all continuous functions on a compact Hausdorff space. Therefore every bounded F-ring is a semisimple real Banach algebra. "Real" is used here in the classical sense, that is, a partially ordered ring R is