NILPOTENT, ALGEBRAIC AND QUASI-REGULAR ELEMENTS IN RINGS AND ALGEBRAS

作者：Stopar Nik^*

来源：Proceedings of the Edinburgh Mathematical Society, 2017, 60(3): 753-769.

DOI：10.1017/S0013091516000353

• pi-algebraic element
• nil ring
• integral ring
• quasi-regular element
• Jacobson radical
• upper nilradical

We prove that an integral Jacobson radical ring is always nil, which extends a well-known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p(x) with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Kothe conjecture: namely, the integral rings.

• 出版日期2017-8