abstract algebra – What does the notation $mathbb{Z}_q[X]/(X^n+1)$ mean and how is multiplication defined there?

I recently encountered the notation which is referred to as ring (in the description of a cryptography scheme, Dilithium, here, page 4):

(…) The key generation algorithm generates a $k times l$ matrix $bf{A}$ each of whose entries is a polynomial in the ring $R_q = mathbb{Z}_q/(X^n+1)$

I am not entirely sure how to read this notation:

  • Is this simply the polynomials of maximum degree where all coefficients are in $Z_q$?
  • How is the multiplication of two polynomials defined in this ring?
  • Does the matrix $bf{A}$ now consist of $l$ polynomials or $k times l$ polynomials?