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Irreducible Module


A nonzero module M over a ring R whose only submodules are the module itself and the zero module. It is also called a simple module, and in fact this is the name more frequently used nowadays (Rowen, 1988). Behrens' (1972, p. 23) definition includes the additional condition that RM be not the zero module.

Sometimes, the term irreducible is used as an abbreviation for meet-irreducible (Kasch 1982), which means that the intersection of two nonzero submodules is always nonzero.

These two irreducibility notions are different: every irreducible module is meet-irreducible, but the converse does not hold. For example, the submodules of Z_4 are {0^_}, Z_4 and {0^_,2^_}, so Z_4 is not irreducible, whereas it is certainly meet-irreducible.

This ambiguity in terminology is solved in the context of rings, since a simple ring is a ring that is irreducible as a module over itself, whereas an irreducible ring is a ring which is meet-irreducible as a module over itself.

Irreducible modules play a crucial role in representation theory.


See also

Irreducible Ring

This entry contributed by Margherita Barile

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References

Behrens, E.-A. Ring Theory. New York: Academic Press, p. 23, 1972.Kasch, F. Modules and Rings. London, England: Academic Press, p. 161, 1982.Rowen, L. "Simple Rings and Modules." in Ring Theory, Vol. 1. San Diego, CA: Academic Press, pp. 19-21, 1988.

Referenced on Wolfram|Alpha

Irreducible Module

Cite this as:

Barile, Margherita. "Irreducible Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/IrreducibleModule.html

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