query: Union of partial ring is not a partial ring
positive_passages.docid: 22245
positive_passages.text: Show the union of two subrings is generally not a subring
negative_passages.docid: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 95373
negative_passages.text: Additive set function on a semiring of sets
negative_passages.docid: 63338
negative_passages.text: Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
negative_passages.docid: 91021
negative_passages.text: Torsion elements of a group aren't necessarily a subgroup
negative_passages.docid: 46843
negative_passages.text: A ring without the Invariant Basis Number property
negative_passages.docid: 64871
negative_passages.text: Closure of Union contains Union of Closures
negative_passages.docid: 105887
negative_passages.text: Ring and Subring with different Identities
negative_passages.docid: 27899
negative_passages.text: Can we prove the existence of $A\cup B$ without the union axiom?
negative_passages.docid: 35222
negative_passages.text: Characterizing units in polynomial rings
negative_passages.docid: 17059
negative_passages.text: A ring element with a left inverse but no right inverse?
negative_passages.docid: 70538
negative_passages.text: Intersection and union of simply connected domain
negative_passages.docid: 92420
negative_passages.text: Characteristic of  a Non-unital Integral Ring
negative_passages.docid: 30465
negative_passages.text: Determining which rings have a module which is not semisimple
negative_passages.docid: 4507
negative_passages.text: Localization of a ring which is not a domain
negative_passages.docid: 5360
negative_passages.text: Ideal contained in a finite union of prime ideals
negative_passages.docid: 70861
negative_passages.text: $\mathbb P^n\times \mathbb P^m\not\cong \mathbb P^{m+n}$