Obtain vector of non-zero elements in sparse matrix, keeping both column and row names

Suppose I have the following matrix:

mat <- matrix(data = c(1, 2, 3, 0, 0, 0, 0, 0, 0, 
                       0, 0, 0, 2, 3, 4, 0, 0, 0,
                       0, 0, 0, 0, 0, 0, 5, 6, 7),
              nrow = 9, 
              dimnames = list(c(paste0("x", 1:3),
                                paste0("y", 1:3),
                                paste0("z", 1:3)),
                              c("a", "b", "c")))

   a b c
x1 1 0 0
x2 2 0 0
x3 3 0 0
y1 0 2 0
y2 0 3 0
y3 0 4 0
z1 0 0 5
z2 0 0 6
z3 0 0 7

Instead of a matrix, I want a vector only keeping the non-zero elements.

red <- apply(mat, 1, function(x) x[x != 0])

x1 x2 x3 y1 y2 y3 z1 z2 z3 
 1  2  3  2  3  4  5  6  7 

Is there a way to have the reduced vector keep the column names as well? Preferably the pattern "colname + character + rowname". See desired output below. Note that we do not know in advance how many columns/rows there will be, nor how they are named.

a=x1 a=x2 a=x3 b=y1 b=y2 b=y3 c=z1 c=z2 c=z3 
 1    2    3    2    3    4    5    6    7 

Thank you in advance!

>Solution :

A base solution with which(..., arr.ind = TRUE):

ind <- which(mat != 0, arr.ind = TRUE)
setNames(
  mat[ind],
  paste(colnames(mat)[ind[, 'col']], rownames(mat)[ind[, 'row']], sep = '=')
)

# a=x1 a=x2 a=x3 b=y1 b=y2 b=y3 c=z1 c=z2 c=z3 
#    1    2    3    2    3    4    5    6    7