It is shown that if A is an object in an exactly definable category \({\mathcal {C}}\) such that A has finite pure Goldie dimension and that every pure monomorphism \(A\rightarrow A\) is an isomorphism, then its endomorphism ring \(\hbox {End}_{{\mathcal {C}}}(A)\) is semilocal. Also, it is proved that every subobject of a pure quotient finite dimensional pure injective object of an exactly definable category has a semilocal endomorphism ring.

中文翻译：

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完全可定义类别的纯Goldie尺寸

结果表明，如果A是完全可定义类别\（{\ mathcal {C}} \\）中的对象，使得A具有有限的纯Goldie维数，并且每个纯单态\（A \ rightarrow A \）都是同构，那么它的同态环\（\ hbox {End} _ {{\ mathcal {C}}}（A）\）是半局部的。同样，证明了可精确定义类别的纯商有限维纯内射对象的每个子对象都有一个半局部同态环。