query: Prove: A nonempty subset B of a ring A is closed with respect to addition and negatives iff B is closed with respect to subtraction
positive_passages.docid: 142739
positive_passages.text: Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
negative_passages.docid: 47434
negative_passages.text: Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
negative_passages.docid: 88307
negative_passages.text: Commutative property of ring addition
negative_passages.docid: 130073
negative_passages.text: $N(ab) \geq N(a)$ is not necessary for the ring to be euclidean
negative_passages.docid: 150738
negative_passages.text: Proof of $(A - B) - C = A - (B \cup C)$
negative_passages.docid: 75966
negative_passages.text: Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
negative_passages.docid: 25710
negative_passages.text: $A ⊂ B$ if and only if $A − B = ∅$
negative_passages.docid: 43310
negative_passages.text: Proving $A\cap(B-C)$ is equal to $(A\cap B)-(A\cap C)$
negative_passages.docid: 115606
negative_passages.text: Proofs involving $ (A\setminus B) \cup (A \cap B)$
negative_passages.docid: 19838
negative_passages.text: Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
negative_passages.docid: 87078
negative_passages.text: Prove the following set is an ideal of the ring $Z[x]$
negative_passages.docid: 21229
negative_passages.text: If an element is not in the set difference $A\setminus B \: \:?$
negative_passages.docid: 117590
negative_passages.text: Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
negative_passages.docid: 106894
negative_passages.text: If a subring of a ring R has identity, does R also have the identity?
negative_passages.docid: 116009
negative_passages.text: Proving that if a set is both open and closed then it is equal to the real numbers
negative_passages.docid: 83100
negative_passages.text: Prove that the center of a ring is a subring.
negative_passages.docid: 91962
negative_passages.text: In a ring $a*a=0$ then $a+a=0$