query: How to show that the ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$ is finite without using the concepts of vector space?
positive_passages.docid: 19112
positive_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 17748
negative_passages.text: Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
negative_passages.docid: 69353
negative_passages.text: Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite?
negative_passages.docid: 92822
negative_passages.text: Finite number of elements generating the unit ideal of a commutative ring
negative_passages.docid: 76749
negative_passages.text: Example of a finite non-commutative ring without a unity
negative_passages.docid: 57310
negative_passages.text: Prove $\mathbb{Z}$ is not a vector space over a field
negative_passages.docid: 100340
negative_passages.text: Invariant dimension property on a ring without zero divisors
negative_passages.docid: 93474
negative_passages.text: A Finite Boolean Ring is Generated by Finitely Many Copies of $\mathbb{Z} / 2 \mathbb{Z}$
negative_passages.docid: 38028
negative_passages.text: finitely generated k-algebra and polynomial ring
negative_passages.docid: 19310
negative_passages.text: Identify the ring $\mathbb{Z}[x]/(2x)$
negative_passages.docid: 74974
negative_passages.text: When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module?
negative_passages.docid: 84149
negative_passages.text: Prove that an infinite ring with finite quotient rings is an integral domain
negative_passages.docid: 85987
negative_passages.text: Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain
negative_passages.docid: 21840
negative_passages.text: Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
negative_passages.docid: 151425
negative_passages.text: Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?
negative_passages.docid: 39049
negative_passages.text: Ring With Unity As a Direct Sum of Non-Zero Ideals
negative_passages.docid: 46843
negative_passages.text: A ring without the Invariant Basis Number property