Let p ≥ 3 be a prime, e ≥ 2 an integer and $\mathbb {Z}_{p^{e}}$ the residue ring modulo pe. Let σ(x) be a primitive polynomial of degree n over $\mathbb {Z}_{p^{e}}$ and let G′(σ(x), pe) be the set of primitive linear recurring sequences over $\mathbb {Z}_{p^{e}}$ generated by σ(x). A compressing mapping $\varphi :\mathbb {Z}_{p^{e}}\rightarrow \mathscr {A}$ naturally induces a mapping $\widehat {\varphi }$ on G ′(σ(x), pe), i.e., $\widehat {\varphi }$ maps a sequence (…,si− 1,si,si+ 1,… ) to (…,φ(si− 1),φ(si),φ(si+ 1),… ). For any pair of sequences in $\{\widehat {\varphi }(\underline {s}):\underline {s}\in G^{\prime }_{~}(\sigma (x),p^{e})\}$ , it is desirable to determine whether (at least) one element of $\mathscr {A}$ is distributed differently in them. For $\emptyset \neq D\subseteq \mathscr {A}$ , $\widehat {\varphi }$ is said to be injective on G ′(σ(x), pe) w.r.t. D-uniformity if for any two distinct sequences $\underline {u},\underline {v}\in G^{\prime }_{~}(\sigma (x),p^{e})$ , the distribution of at least one element of D in $\widehat {\varphi }(\underline {u})$ differs from that in $\widehat {\varphi }(\underline {v})$ . A sufficient condition on φ is given to ensure that $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. If $\left (\left ((x^{p^{n}-1}-1)^{2}\bmod \sigma (x)\right ) \bmod p^{3}\right ) \notin p^{2}\mathbb {Z}_{p}$ , then an equivalent condition on φ is obtained to decide whether $\widehat {\varphi }$ is injective on G ′(σ(x), pe) w.r.t. D-uniformity. Furthermore, quantitative estimation suggests that almost all mappings on $\mathbb {Z}_{p^{e}}$ induce injective mappings on G ′(σ(x), pe) as p and e increase.