The Role of the Jacobson Radical in Defining the Structure of Prime and Maximal Ideals in Artinian Rings

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@article{1724183,
      author = {Quinter Murokoyo, Makila Patrick, Linet Muhati},
      title = {The Role of the Jacobson Radical in Defining the Structure of Prime and Maximal Ideals in Artinian Rings},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {10},
      number = {4},
      pages = {3357-3366},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1724183.pdf},
      abstract = {The Jacobson radical J(R) of a ring R is a fundamental tool in the structural analysis of Artinian rings. This paper examines how J(R) governs the behaviour of prime and maximal ideals in such rings. The basic properties of the radical are established, including the unit characterisation, its behaviour under quotients and its relation to simple modules, together with Nakayama’s Lemma and the computation of the radical for matrix rings, direct products and commutative rings. The central finiteness result is then proved: in a left Artinian ring the Jacobson radical is nilpotent. From this it follows that the quotient R/J(R) is semisimple Artinian, and a finite product of fields when R is commutative. These results are used to prove that every prime ideal of a commutative Artinian ring is maximal, and that the Jacobson radical coincides with the nilradical in that setting, with the noncommutative analogue identifying J(R) as the largest nilpotent left ideal of a left Artinian ring. Examples are given throughout, including instances establishing that the Artinian hypothesis is sufficient but not necessary for the identification of the radical with the nilradical, and that it cannot be dropped altogether.},
      keywords = {Artinian ring; Jacobson radical; maximal ideal; nilpotent ideal; nilradical; prime ideal; semisimple ring},
      month = {October},
  }