Category theory MOC

Simple object

Let 𝖢 be a category. An object 𝑋 ∈𝖢 is called simple iff its only quotients (in the sense of coëqualizers) are the terminal object and 𝑋 itself.¹ #m/def/cat If 𝖢 is abelian category, it is sufficient for 𝑋 to have no subobjects.

Examples

• Simple group in 𝖦𝗋𝗉.
• Simple module in 𝑅𝖬𝗈𝖽.
• Simple ring in 𝖱𝗂𝗇𝗀.

Properties

• Schur's lemma in abelian categories

See also

• Semisimple object in an abelian category

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