query: If the Additive Subgroup of a Ring is Cyclic, is the Multiplicative Identity Necessarily a Generator?
positive_passages.docid: 18361
positive_passages.text: Does the unit generate the additive group in a unital ring with cyclic additive group?
negative_passages.docid: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 88307
negative_passages.text: Commutative property of ring addition
negative_passages.docid: 14255
negative_passages.text: Commutator subgroup and cyclic normal subgroup
negative_passages.docid: 126590
negative_passages.text: Why is the multiplicative group of a finite field cyclic?
negative_passages.docid: 75986
negative_passages.text: If all Subgroups are Cyclic, is group Cylic?
negative_passages.docid: 25744
negative_passages.text: Multiplication in Permutation Groups Written in Cyclic Notation
negative_passages.docid: 2304
negative_passages.text: What is the quotient of a cyclic group of order $n$ by a cyclic subgroup of order $m$?
negative_passages.docid: 154959
negative_passages.text: Proof of associativity of GCD?
negative_passages.docid: 158873
negative_passages.text: Is every group the unit group of some ring?
negative_passages.docid: 114187
negative_passages.text: In a finite cyclic group, the generator has the order of the group
negative_passages.docid: 12110
negative_passages.text: How are $2\mathbb{Z}\ncong3\mathbb{Z}$ different as rings?
negative_passages.docid: 99032
negative_passages.text: Prove that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.
negative_passages.docid: 76446
negative_passages.text: Why are cyclic groups of order $p$ (prime) unique up to isomorphism?
negative_passages.docid: 145561
negative_passages.text: About the additive group and the multiplicative group of a field
negative_passages.docid: 145112
negative_passages.text: Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
negative_passages.docid: 58397
negative_passages.text: Every cyclic group with order > 2 has at least 2 distinct generators