query: Let $R$ be a commutative ring and $a_1, \dots, a_n$ elements of $R$. Prove that $\frac{R[x_1, \dots, x_n]}{(x_1-a_1, \dots, x_n-a_n)} \cong R.$
positive_passages.docid: 30327
positive_passages.text: $\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
negative_passages.docid: 77198
negative_passages.text: Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
negative_passages.docid: 105363
negative_passages.text: $\mathbb Z^n/\langle (a,...,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
negative_passages.docid: 464
negative_passages.text: Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
negative_passages.docid: 1425
negative_passages.text: The Chinese Remainder Theorem for Rings.
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: 98818
negative_passages.text: Quotient ring of a localization of a ring
negative_passages.docid: 38946
negative_passages.text: Show that $\{\cup_{n\in K} (n, n+1]: K \subset \mathbb{Z}\}$ is a $\sigma$-algebra on $\mathbb{R}$
negative_passages.docid: 42780
negative_passages.text: Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
negative_passages.docid: 51898
negative_passages.text: Show that $\operatorname{Hom}_{R} (R,M) \cong M$
negative_passages.docid: 35777
negative_passages.text: $\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$
negative_passages.docid: 121156
negative_passages.text: Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
negative_passages.docid: 90438
negative_passages.text: To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
negative_passages.docid: 88307
negative_passages.text: Commutative property of ring addition
negative_passages.docid: 16023
negative_passages.text: Are associates unit multiples in a commutative ring with $1$?
negative_passages.docid: 6896
negative_passages.text: If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
negative_passages.docid: 82591
negative_passages.text: Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.