caa a22 4500 605524416 CHVBK 20210128100754.0 cr unu---uuuuu 210128e20150501xx s 000 0 eng 10.1007/s10958-015-2329-1 doi (NATIONALLICENCE)springer-10.1007/s10958-015-2329-1 Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals [Elektronische Daten] [E. da Costa, A. Krasilnikov] Let K be a field and let ℕ = {1, 2, . . .} be the set of all positive integers. Let R n = K[x ij | 1 ≤ i ≤ n, j ∈ ℕ] be the ring of polynomials in x ij (1 ≤ i ≤ n, j ∈ ℕ) over K. Let S n = Sym({1, 2, . . . , n}) and Sym(ℕ) be the groups of permutations of the sets {1, 2, . . . , n} and ℕ, respectively. Then S n and Sym(ℕ) act on R n in a natural way: τ (x ij ) = x τ(i)j and σ(x ij ) = x iσ(j), for all i ∈ {1, 2, . . . , n} and j ∈ ℕ, τ ∈ S n and σ ∈ Sym(ℕ). Let R ¯ $$ \overline{R} $$ n be the subalgebra of (S n -)symmetric polynomials in R n , i.e., R ¯ n = f ∈ R n τ f = f f o r each τ ∈ S n . $$ {\overline{R}}_n=\left\{f\in {R}_n\left|\tau (f)=f\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{each}\ \tau \right.\in {\mathrm{S}}_n\right\}. $$ An ideal I in R ¯ $$ \overline{R} $$ n is called Sym(ℕ)-invariant if σ(I) = I for each σ ∈ Sym(ℕ). In 1992, the second author proved that if char(K) = 0 or char(K) = p > n, then every Sym(ℕ)-invariant ideal in R ¯ $$ \overline{R} $$ n is finitely generated (as such). In this note, we prove that this is not the case if char(K) = p ≤ n. We also survey some results on Sym(ℕ)-invariant ideals in polynomial algebras and some related topics. Springer Science+Business Media New York, 2015 da Costa E. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil aut Krasilnikov A. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil aut Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 206/5(2015-05-01), 505-510 1072-3374 206:5<505 2015 206 10958 https://doi.org/10.1007/s10958-015-2329-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2329-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- da Costa E. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil aut NATIONALLICENCE 700 1- Krasilnikov A. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 206/5(2015-05-01), 505-510 1072-3374 206:5<505 2015 206 10958