query: If $R_1,..$ are rings with identity and $I$ is an ideal in $R_1\times.. R_n$, then $I=A_1\times...\times A_m$,where each $A_i$ is an ideal in $R_i$.
positive_passages.docid: 29019
positive_passages.text: Ideals in direct product of rings
negative_passages.docid: 16023
negative_passages.text: Are associates unit multiples in a commutative ring with $1$?
negative_passages.docid: 24259
negative_passages.text: Quotient ring of $T/I$
negative_passages.docid: 72507
negative_passages.text: Does $ab=1$ imply $ba=1$ in a ring?
negative_passages.docid: 37393
negative_passages.text: $R$ is commutative, $I$,$J$ are ideals, $I+J=R$, then $IJ=I\cap J$
negative_passages.docid: 65224
negative_passages.text: Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
negative_passages.docid: 58640
negative_passages.text: $R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$
negative_passages.docid: 43990
negative_passages.text: R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
negative_passages.docid: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 30714
negative_passages.text: A question on ring homomorphisms and maximal ideals.
negative_passages.docid: 136654
negative_passages.text: $(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$
negative_passages.docid: 110984
negative_passages.text: Ideals and prime ideals in a commutative ring.
negative_passages.docid: 82953
negative_passages.text: In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$
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: 92583
negative_passages.text: Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
negative_passages.docid: 62917
negative_passages.text: Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
negative_passages.docid: 50445
negative_passages.text: Explaining the product of two ideals