query: $f$ is a unit in $R$ if and only if $a_0$ is a unit and $a_n, ..., a_1$ are nilpotent
positive_passages.docid: 35222
positive_passages.text: Characterizing units in polynomial rings
negative_passages.docid: 60520
negative_passages.text: Upper bound for $|f(0)|$ based on its behavior within the unit disk.
negative_passages.docid: 148550
negative_passages.text: Units in quotient ring of $\mathbb Z[X]$
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
negative_passages.docid: 14403
negative_passages.text: If $n = p_1^{a_1} ... p_r^{a_r}$ the set of nilpotents of $\mathbb{Z}_n$ is $(p_1 p_2 ... p_{r-1} p_r)\Bbb Z_n$
negative_passages.docid: 103830
negative_passages.text: Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
negative_passages.docid: 47709
negative_passages.text: If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units
negative_passages.docid: 115273
negative_passages.text: Nilpotent matrix and basis for $F^n$
negative_passages.docid: 54479
negative_passages.text: If $x,y$ are nilpotent and commute, $x+y $ is nilpotent.
negative_passages.docid: 64077
negative_passages.text: $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.
negative_passages.docid: 133758
negative_passages.text: Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.
negative_passages.docid: 64484
negative_passages.text: Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$
negative_passages.docid: 37867
negative_passages.text: If $0$ is the only eigenvalue of a linear operator, is the operator nilpotent
negative_passages.docid: 98756
negative_passages.text: Unit ball in $C[0,1]$ not sequentially compact
negative_passages.docid: 56097
negative_passages.text: If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
negative_passages.docid: 39788
negative_passages.text: $a+1,a-1$ invertible for nilpotent element
negative_passages.docid: 128043
negative_passages.text: $A^n=0$ for $A$ nilpotent without the use of minimal polynomial