On relation between statistical ideal and ideal generated by a modulus function

Ideal on an arbitrary non-empty set $\Omega$ it's a family of subset $\mathfrak{I}$ the which satisfies following axioms: $\Omega \notin \mathfrak{I}$, if $A, B \in then $A \cup \mathfrak{I}$ and $D \subset A$, \mathfrak{I}$. The ideal theory is very popular branch modern mathematical research. In our paper we study some classes ideals all positive integers $\mathbb{N}$, namely statistical convergence $\mathfrak{I}_s$ $\mathfrak{I}_f$ generated by modular function $f$. Statistical subsets $\mathbb{N}$ whose natural density equal to 0, i.e. \mathfrak{I}_s$ only $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{\#\{k \leq n: k A\}}{n} = 0$. A $f:\mathbb{R}^+ \mathbb{R}^+$ called function, $f(x) 0$ $x 0$, $f(x + y) f(x) f(y)$ for $x, y \in\mathbb{R}^+$, \le whenever y$, $f$ continuous from right finally $\lim\limits_{n \infty} f(n) \infty$. Ideal, with zero $f$-density, in other words, \mathfrak{I}_f$ \infty}\frac{f(\#\{k A\})}{f(n)} It known that true: $\mathfrak{I}_f \mathfrak{I}_s$. research give complete description those functions Then analyse obtained result, partial cases it prove one simple sufficient condition equality last section this article devoted examples modulus $f, g$ $\mathfrak{I}_g \neq Namely, x^p$ where $p (0, 1]$ have \mathfrak{I}_s$; $g(x) \log(1 x)$, obtain consider more complicated given recursively demonstrate conditions main theorem can't be reduced mentioned above.