Left Artinian Algebraic Algebras

• 来源期刊: 代数集刊（英文版）
• 学科: 数学
• 关键词: division rings semisimple rings algebraic algebras artinian rings
• 年，卷（期）: 2001,（4）
• 页码范围: 463-470
• 页数: 8页
• 分类号: O1
• 语种: 英文

摘要

Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.