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In this chapter, we develop the basic theory of non-commutative rings, putting special attention on the Jacobson radical. Let M be an R-module and S a subset of M. The annihilator of S is \(\begin{aligned} \operatorname {Ann}_R(S)=\{ a\in R:as=0\;\text {for all } s\in S\}. \end{aligned}\) If \(S=\{m\}\) , then we will write \(\operatorname {Ann}_R(m)=\operatorname {Ann}_R(\{m\})\) . It is an exercise to check that \(\operatorname {Ann}_R(S)\) is a left ideal of R. Furthermore, if S is a submodule of M, then \(\operatorname {Ann}_R(S)\) is an ideal of R.

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The Jacobson Radical

  • 章节

  • 发表日期：

    2026年

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  • Ferran Cedó,
  • Leandro Vendramin

摘要

In this chapter, we develop the basic theory of non-commutative rings, putting special attention on the Jacobson radical. Let M be an R-module and S a subset of M. The annihilator of S is \(\begin{aligned} \operatorname {Ann}_R(S)=\{ a\in R:as=0\;\text {for all } s\in S\}. \end{aligned}\) If \(S=\{m\}\) , then we will write \(\operatorname {Ann}_R(m)=\operatorname {Ann}_R(\{m\})\) . It is an exercise to check that \(\operatorname {Ann}_R(S)\) is a left ideal of R. Furthermore, if S is a submodule of M, then \(\operatorname {Ann}_R(S)\) is an ideal of R.