TY - JOUR

T1 - On the existence and non-existence of complementary radical and semisimple classes

AU - Veldsman, S.

AU - Wiegandt, R.

PY - 1984

Y1 - 1984

N2 - In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.

AB - In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.

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U2 - 10.1080/16073606.1984.9632332

DO - 10.1080/16073606.1984.9632332

M3 - Article

AN - SCOPUS:84952196889

SN - 1607-3606

VL - 7

SP - 213

EP - 224

JO - Quaestiones Mathematicae

JF - Quaestiones Mathematicae

IS - 3

ER -