query: $S^{-1}R\cong \frac{R[x]}{(sx-1)}$ for $S = \langle s^k\rangle\ $ [fraction ring as quotient ring]
positive_passages.docid: 26644
positive_passages.text: Does this give a localization of a ring?
negative_passages.docid: 50241
negative_passages.text: Cardinality of the quotient ring $\mathbb Z_5[i]/\langle 1+i\rangle$
negative_passages.docid: 128710
negative_passages.text: Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$
negative_passages.docid: 87823
negative_passages.text: GAP: how to work with elements of the group ring of the symmetric group $\mathbb{C}[S_k]$?
negative_passages.docid: 87946
negative_passages.text: Proving that the unities of a ring form a group under multiplication
negative_passages.docid: 105913
negative_passages.text: Calculations in quotient rings
negative_passages.docid: 7011
negative_passages.text: Integral domain with fraction field equal to $\mathbb{R}$
negative_passages.docid: 18036
negative_passages.text: Localization by a non-nilpotent as a polynomial quotient
negative_passages.docid: 53266
negative_passages.text: Show that $R/I\otimes_R R/J\cong R/(I+J)$
negative_passages.docid: 2907
negative_passages.text: How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$?
negative_passages.docid: 157840
negative_passages.text: Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?
negative_passages.docid: 86717
negative_passages.text: Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$
negative_passages.docid: 88199
negative_passages.text: The commutator subgroup of a quotient in terms of the commutator subgroup and the kernel
negative_passages.docid: 152297
negative_passages.text: Understanding quotients of $\mathbb{Q}[x]$
negative_passages.docid: 94579
negative_passages.text: $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$
negative_passages.docid: 155445
negative_passages.text: What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
negative_passages.docid: 83050
negative_passages.text: Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.