"This is his household."
Translation:Das ist sein Haushalt.
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Why is "Dieses ist sein Haushalt." incorrect? A brief search turned up https://en.wiktionary.org/wiki/dies#German, which indicates that Dies and Dieses should be interchangeable. Is this not correct?
The possessive determiners mein, dein, sein, ihr, unser, euer act like the indefinite articles ein, kein in taking no ending for masculine nominative and neuter nominative/accusative.
So for masculine nominative, it's sein, not *seiner (which would be only feminine genitive/dative or plural dative).
Also, you wrote:
since there is no definite article
Firstly, that rule is for adjectives; secondly, the criterion is not "definite article" but rather "determiner that shows an ending for gender/number/case".
Definite articles are just one form of determiner; other possibilities include indefinite article (eine), demonstrative determiners (diese), and possessive determiners (meine).