simple ring

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simple ring

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simple ring (plural simple rings)

1. (algebra, ring theory) A ring that contains no nontrivial ideals (i.e., no (two-sided) ideals other than the zero ideal and the ring itself).
   • 1956, Nathan Jacobson, Structure of Rings, American Mathematical Society, page 43:

     Theorem 2. A semi-simple ring ${\displaystyle {\mathfrak {A}}}$ satisfying the minimum condition can be decomposed in only one way as a direct sum of ideals which are simple rings.

   • 1969, Frederick Michael Hall, An Introduction to Abstract Algebra, volume 2, Cambridge University Press, page 195:

     By theorem 7.7.1 any field is a simple ring.

   • 1994, P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic Abstract Algebra, 2nd edition, Cambridge University Press, page 204:

     A field is clearly a simple ring. Indeed, a commutative simple ring with unity must be a field (Problem 1, Section 1). An example of a noncommutative simple ring is F_n, the n × n matrix ring over a field F, n > 1.

   • 2017, Ramji Lal, Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group Extensions and Schur Multiplier, Springer, page 335:

     Since ${\displaystyle M_{n}(D)}$ has no nonzero proper two-sided ideals, it follows from the above discussion that ${\displaystyle M_{n}(D)}$ is a left simple ring. We shall see that every simple ring is isomorphic to ${\displaystyle M_{n}(D)}$ for some ${\displaystyle n}$, and for some division ring ${\displaystyle D}$.

• local ring

• field

• semisimple ring
• simple module
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