query: Why is the group ring just a ring, not a field?
positive_passages.docid: 26483
positive_passages.text: Group algebra ever a field?
negative_passages.docid: 121643
negative_passages.text: Why isn't the zero ring the field with one element?
negative_passages.docid: 124853
negative_passages.text: Why does "dimension" of the generator matter for cohomology rings?
negative_passages.docid: 28327
negative_passages.text: (Simple) Examples on Non-Commutative Rings
negative_passages.docid: 39369
negative_passages.text: Why are there rings for Dwarves and Elves and Men, but not Hobbits or Orcs?
negative_passages.docid: 55769
negative_passages.text: Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers?
negative_passages.docid: 79641
negative_passages.text: Why are rings called rings?
negative_passages.docid: 76813
negative_passages.text: The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
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: 16643
negative_passages.text: A slick proof that a field which is finitely generated as a ring is finite
negative_passages.docid: 17059
negative_passages.text: A ring element with a left inverse but no right inverse?
negative_passages.docid: 138383
negative_passages.text: group of units in a topological ring
negative_passages.docid: 2458
negative_passages.text: Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?
negative_passages.docid: 52212
negative_passages.text: If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
negative_passages.docid: 124455
negative_passages.text: Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
negative_passages.docid: 97751
negative_passages.text: Center of a group ring?
negative_passages.docid: 91622
negative_passages.text: Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.