On maximal ideals and the brown-mccoy radical of polynomial rings
On maximal ideals and the brown-mccoy radical of polynomial rings
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论极大理想与多项式环的布朗-麦考伊根
DOI：
10.1080/00927879808826292
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发表时间：
1998
影响因子：
0.7
通讯作者：
A. Smoktunowicz
中科院分区：
文献类型：
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作者：
E. Puczyłowski;A. Smoktunowicz
本文得到了多项式环R[x]在一个不定式x上的极大理想的一些结果，特别是用一个恒等式完成了环R的Ferrero刻划[5]，使得R[x]包含一个极大R-不相交理想，即一个满足M n R = 0的极大理想M。我们还得到了R[x]的Brown-McCoy自由基的几个结果。回想一下，对于给定环a， a的Brown-McCoy根U(a)被定义为a的所有理想I的交集，使得a /I是一个具有单位元的简单环。特别地，一个环是Brown-McCoy基当且仅当它不能同态映射到具有单位元的环上，或等价地映射到具有单位元的简单环上。在2010年，Krernpa证明了对于每个环R， U(R[xJ]) = (U(R[x]) n R)[xJ]。我们将证明U(R[x]) n R等于R的所有素数理想I的交点，使得R/I的中心与RII的每个非零理想有一个非零交点。特别地，如果R是一个零环，则R[x]是Brown-McCoy基，即R[x]不能同态映射到上
In this paper we obtain some results on maximal ideals of polynomial rings R[x] in one indeterminate x. In particular we complete Ferrero's characterization [5] of rings R with an identity such that R[x] contains a maximal R-disjoint ideal, i.e., a maximal ideal M satisfying M n R = 0. We also get several results on the Brown-McCoy radical of R[x]. Recall that for a given ring A the Brown-McCoy radical U(A) of A is defined as the intersection of all ideals I of A such that A/I is a simple ring with an identity. In particular a ring is Brown-McCoy radical if and only if it cannot be homomorphically mapped onto a ring with an identity, or equivalently, onto a simple ring with an identity. In [7] Krernpa proved that for every ring R, U(R[xJ) = (U(R[x]) n R)[xJ. We shall show that U(R[x]) n R is equal to the intersection of all prime ideals I of R such that the centre of R/I has a non-zero intersection with each non-zero ideal of RII. In particular, if R is a nil ring, then R[x] is Brown-McCoy radical, i.e., R[x] cannot be homomorphically mapped onto